Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Finding ranks and nullities of linear maps

I am confused about ranks, nullities and bases of the kernel. From what I understand the rank is the dimension of a vector space generated by a matrix. How would I do the following examples? Find ...
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1answer
20 views

Suppose that $A \in M_{m\times n}$ & $B, C \in M_{n\times m}$ are matrices that satisfy $BA= I_n$ and $AC=I_m$. Prove that $B=C$.

Suppose that $A \in M_{m\times n}$ & $B, C \in M_{n\times m}$ are matrices that satisfy $BA= I_n$ and $AC=I_m$. Prove that $B=C$. In my mind, a good way to go about this proof is proving that ...
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0answers
12 views

Hyperplane of an mn-dimensional space [on hold]

Can someone explain to me why the hyperplane of a $mn$-dimensional space would have dimension $(m-1)n$?
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0answers
37 views

Differentiating a matrix product

In one of the books I found that given that for a linear system $x'=Ax$, there exists a matrix $Q:=\int\limits_0^\infty B(t)dt$, where $B(t)=e^{tA^T}e^{tA}$, and $V(x) = x^T Q x$, ...
0
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2answers
22 views

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}$. I know that vectors that are orthogonal will have a dot product of 0. So here's what I was thinking: ...
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1answer
28 views

Comparing matrix norm with the norm of the inverse matrix

I need help understanding and solving this problem. Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$ Is this just asking me to get the magnitude of ...
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1answer
25 views

Determinant of a tuple of vectors: is this a thing? If so, where can I learn more?

Let $k \leq n$ denote a pair of fixed but arbitrary natural numbers. Definition 0. Write $\varphi$ for the unique $\mathbb{R}$-linear function $$\Lambda^k\mathbb{R}^n \rightarrow \mathbb{R}$$ such ...
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2answers
23 views

Finding a linear transformation with a given null space

The problem statement is, Find a linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satisfying $4x_1-3x_2+x_3=0$ is the (i) null space of $T$ (ii) range of $T$. ...
0
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1answer
23 views

Relating regression to projection?

I recently learned that one can think of regression as a projection of a vector in a high dimension space onto the other vector. I tried implementing this and got it to work: ...
2
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3answers
61 views

Why do I have to show this subspace is an invariant subspace?

Consider a vector space $V \cong \mathbb{R}^n$ with an operator $I \in O(n)$ satisfying the property $I^2 = -Id_{V}$. See Linear Complex Structure for context. I want to show that $V$ has real ...
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2answers
35 views

Show that an inverse of a bijective linear map is a linear map.

So I've got a bijection. It clearly has an inverse, but how exactly do I prove that the inverse is a linear map as well? enter image description here
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1answer
24 views

Dimension of the subspace of a vector space spanned by the following vectors.

I know that in order to find a subsequence that is a basis of a subspace is to check whether the given vectors are linearly independent and whether they span the subspace. However how can I find the ...
0
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1answer
17 views

A question concerning isometries and determinants

Let $e_1 = (1, 0)$ and $e_2 = (0, 1)$. Assume that $f : \Bbb R^2 → \Bbb R^2$ is an isometry of the plane fixing $(0, 0)$. Let $f(e_1) = (a, b)$ and $f(e_2) = (c, d)$, and let $A = \begin{vmatrix} ...
0
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1answer
20 views

What's the period of this matrix?

Consider the matrix $$ A = \begin{pmatrix} 0.1 & 0.3 & 0.4 & 0.2 \\ 0.2 & 0.4 & 0.0 & 0.4 \\ 0.0 & 0.3 & 0.5 & 0.2 \\ 0.5 & 0.3 & 0.2 & 0.0 ...
0
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2answers
22 views

Finding the basis of a subspace

I understand that the basis of a subspace defined by this equation requires you to find a combination of $x_1,x_2,x_3$ that satisfy this equation [so $(-1,0,2)$ for example]. But how do you know how ...
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2answers
27 views

Proving that if $A$ is diagonalisable then $\chi_A(A) = 0$

This could be a very simple question to answer, but I'm unsure how to prove this. If you have a diagonalisable matrix $A$, prove that $\chi_A(A)$ is the zero matrix. (where $\chi_A(x)$ is the ...
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3answers
20 views

Question about how the determinant of a square matrix can help determine whether a set of vectors is a basis.

I have a linear algebra midterm tomorrow. While it's highly unlikely a question of this type shows up, I really wanted to understand this because I am curious since I've spent so long without coming ...
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1answer
25 views

Proving that a group representation is *not* a direct sum of irreducible represenations.

Problem Statement: Let $x$ be a generator of a cyclic group $G$ of order $p$. Sending $x\mapsto \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}$ defines a matrix representation $G\rightarrow ...
0
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2answers
23 views

$a+b$ for $ax+3y=5$ and $2x+by=3$

If $ax+3y=5$ and $2x+by=3$ represent the same straight line, then what does a+b equal? I've tried this, $ax+3y=5$ and $2x+by=3$ Multiply to equal 15 so they equal each other ...
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1answer
28 views

$C_o= \{(x_n):x_n \in R, x_n \rightarrow 0 \}$ and $ M=\{(x_n):x_n \in C_0,~~ x_1+ x_2+…+x_{10}=0\}$ then dimension of $(\frac{C_0}{M})$ is equal to

Let $C_o= \{(x_n):x_n \in R, x_n \rightarrow 0 \}$ and $ M=\{(x_n):x_n \in C_0,~~ x_1+ x_2+...+x_{10}=0\}$ then dimension of $(\frac{C_0}{M})$ is equal to
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1answer
11 views

Conditioning in regards to matrix vector product

This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of ...
0
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1answer
23 views

Independence, Inverse and Additive Identity for vector with defined vector addition and scalar multiplication

I am struggling to find my bearings on this question. I am confident that I can do parts a and d. I have no clue how to approach b. I was also wondering if the redefined vector addition and scalar ...
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0answers
30 views

About a particular definition of “tensor”

I came across this quiet new to me way of defining "tensors", That a tensor $A$ is a map of the form, $A : \mathbb{R}^{n \times m_1} \times \mathbb{R}^{n \times m_2} \times .. \times \mathbb{R}^{n ...
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1answer
23 views

How to tell that $W$ is a subspace of $ \mathbb R^3$?

To do this problem, I wrote this matrix in RREF form and found that $V_3$ is $-1V_1 + 2V_2$. This demonstrates that these planes are a basis for $ \mathbb R^2$. However, I am not sure to extend that ...
2
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1answer
23 views

normal operator equation

let $S: V \to V$ linear transformation in a inner product space of $\mathbb{C}$. Prove that $S$ is normal iff $$\|S(x)\| = \|S^*(x)\|$$ That's what I have done so far: if $S$ is normal than $$SS^* ...
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2answers
48 views

Prove that elementary matrices perform row operations

How to prove that elementary matrices actually perform their intended row operations: multiplying by a constant, adding a multiple of one row to another, and switching two rows? I've seen examples ...
3
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1answer
25 views

$A \in SO(3,\mathbb R)\setminus\{I\}$ , then there are exactly two points in $S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$ which are fixed by $A$?

Let $A \in SO(3,\mathbb R)\setminus\{I\}$ , then is it true that there exist exactly two points in $$S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$$ which are fixed by $A$? Or equivalently we ...
3
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2answers
28 views

What is the meaning of the notation [A|B] in Linear Algebra.

I am going through Linear Algebra right now, we are using the book Elementary Linear Algebra by Andrilli. In one of the theorems he uses this notation without really introducing it. Here is the ...
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1answer
22 views

If $A$ is unitary and $f_A(x)=f_B(x)$ and $m_A(x)=m_B(x)$ then $A$ is similar to $B$

Given $A_{n\times n},B_{n\times n} \in \mathbb C$ then: if $A$ is unitary and the characteristic polynomial $f_A(x)=f_B(x)$ then $B$ is also unitary. if $A$ is normal and $f_A(x)=f_B(x)$ ...
1
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1answer
13 views

Write $F$ as a linear combination of elements of $\mathcal B^*$

If $V=\mathbb R[x]_k=\{\sum\limits_{i=1}^ka_ix^i:a_i\in\mathbb R, \forall i\}$ is a vector space of dimension $k+1$ over $K=\mathbb R$ and $\mathcal B=\{1,x,\dots,x^k\}$ is a basis of $V$. The dual ...
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0answers
11 views

How to change a basis of some orthogonal operator to obtain canonical form of operator [on hold]

We have a linear operator $f:X\rightarrow X$ on 3-dimensional real inner product space $X$ which has in O.N. basis $e_1, e_2, e_3$ a matrix $$ \left [ \begin{array}{rrr} \cos t & \sin t & 0\\ ...
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0answers
34 views

Linear Algebra - Transformations, image, kernel [on hold]

Question Define T : R3 → R3 by Tx = (x · (1, 0, −1))(1, 0, −1) + (x · (1, 1, 1))(1, 1, 1) (a) Compute the action of T on the unit vectors i, j, k. (b) Write down the standard ...
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1answer
38 views

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R\{1}. [on hold]

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R \ {1}. (a) Show that a ∗ b ∈ G for all a, b ∈ G. (b) Show that G together with the binary operation G × G → G, (a, b) |→ a ...
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3answers
55 views

The formula $\DeclareMathOperator{tr}{tr}\mathrm{adj}(A)=\tfrac{1}{2}[(\tr A)^2-\tr(A^2)]I_3-[\tr A]A+A^2$ for the adjoint of a $3\times 3$ matrix

Let $A$ be a square matrix of order $3$. Prove that $$ \operatorname{adj}(A) = \tfrac{1}{2} \bigl[ (\operatorname{tr} A)^2 - \operatorname{tr}(A^2) \bigr] I_3 - [\operatorname{tr} A] A + A^2 ...
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2answers
37 views

Describing all the linear transformations satisfying the constraints

How to find the linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satistfying $4x_1-3x_2+x_3=0$ is a) Null space of $T$ b) Range of $T$ I'm not able to ...
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0answers
33 views

Linear Algebra - Transition matrices

Question I have some methodological questions with this exercise: 1. You are given that the transition matric $P_{\mathcal C,\mathcal B}$ from a basis $\mathcal B=\{b_1,\ b_2,\ b_3\}$ to a basis ...
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2answers
27 views

Examine if the set is linearly independent

How do I prove or disprove if $\{1, \cos x, \cos 2x,..., \cos nx\}$ is linearly independent? I tried solving the problem using the definition of linear independence, $\sum_{k=0}^n a_k\cos kx = 0$ ...
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0answers
24 views

Awkwardly formed linear spaces exercise

I came across such an exercise: Let $V$ be a linear space over $K$ such that $\dim V = n$. Show that for any $\alpha_1, \alpha_2, \dots, \alpha_m$ with $ m > n + 1$ there exist $a_1, \dots, ...
3
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4answers
69 views

How to find unknowns $w_1,w_2,w_3$ that satisfy $t=w_1f_1 + w_2f_2 + w_3f_3$?

For any $i \in \{1,2,3\}$, let: $w_i \in [0,1]$ is an unknown number such that $\sum_{i \in \{1,2,3\}} w_i = 1$. $t$ is a known number in $[0,1]$. Suppose that $t = 0.8$. $f_i$ is also a known ...
0
votes
2answers
38 views

how to determine a matrix has a single eigenvalue

Find the jordan form of the matrix $$A = \begin{pmatrix} 1 & 1 & 2 & 2\\ 1 & -2 & -1 & -1\\ -2 & 1 & -1 & -1\\ 1 & 1 & 2 & 2 \\ \end{pmatrix}$$ ...
3
votes
1answer
28 views

Algebra of Linear differential operators, question on Commutativity and Association

The following is a discussion on the following second differential equation $$ \frac{dy^2}{dx} - y = 0 $$ So, let us introduce the following, convention and definition, represent the derivative ...
3
votes
1answer
46 views

Linearity and invertibility of $A^{-1}$

If $A\in L(X)$ then prove that $A^{-1}$ is linear and invertible. Proof: Since $A$ is invertible then $A$ is injective and surjective. We know that $A^{-1}$ defines by $A^{-1}(Ax)=x$. Remark: Also ...
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1answer
26 views

Symmetric Matrix with Positive Eigenvalues

Not all matrix with positive eigenvalues is positive definite, i.e. $\mathbf{x}^\mathsf{T}A\mathbf{x}>0$ for all non zero vector $\mathbf{x}$. For example consider matrix $$A = \begin{bmatrix} 1 ...
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1answer
34 views

Show that $B^TAB$ is symmetric. [on hold]

$A$ is invertible, but it does not say that $A$ is symmetric. By $B^T$ I mean that $B$ is transposed.
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votes
1answer
29 views

Is the the statement is true or false? [on hold]

Suppose $A$ is a $m \times n$ matrix and $V$ is a $m \times 1$ matrix with both $A$ and $V$ having rational entries and suppose the system $AX=V$ has a solution in $\mathbb{R}^n$. Then the equation ...
1
vote
1answer
19 views

If $(A-\lambda I)^{k_j} \vec{v_j} = \vec{0}$ then $(A-\lambda I)\vec{v_j} = V_j$ and $V_j\in \ker(A-\lambda I)^{k_j-1}$

In one book on differential equations and dynamical systems I read that if (1) $(A-\lambda I)^{k_j} \vec{v_j} = \vec{0}$ then (2) $(A-\lambda I)\vec{v_j} = V_j$ and $V_j\in \ker(A-\lambda I)^{k_j-1}$. ...
0
votes
1answer
39 views

Path to Self Study Calculus 1-4 and Linear Algebra [on hold]

For the past year I've taken up self studying mathematics. My initial intent was to study so that when I entered college (currently a junior) I would have most of the basic mathematics for studying ...
1
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1answer
27 views

$U,W$ are subspaces. show $\dim(U+W) = 1+\dim(U \cap W)$, then $\{U+W,U\cap W\}=\{U,W\}$

This is a question from a review package that is causing me some trouble. Let $U,W$ be subspaces of a finite dimensional vector space. Show if $\dim(U+W) = 1+\dim(U \cap W)$, then $\{U+W,U\cap ...
1
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1answer
24 views

Finding an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$

Problem Statement: Let $A=\begin{bmatrix} 2 && 1 \\ 1 && 2 \end{bmatrix}$. Find an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$. I am ...
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2answers
35 views

Finding a Matrix B by knowing its Kernel is the Image of Matrix A

I understand how to find the image($A$). The basis of Im($A$) would be the first two columns of the matrix $A$ (given the two leading 1's in ref are in the first and second columns). So the ...