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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Evaluate a linear system of three equations

Solve for $x, y\ \text{and}\ z\ $: $x-3z=10\\ -x+y+2z=7\\ 2x+2y-5z=-8$ My working: $$\left(\begin{array}{ccc|c} 1 & 0 & -3 & 10 \\ -1 & 1 & 2 & 7 \\ 2 & 2 & -5 & ...
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1answer
76 views

Monomially-equivalent linear codes?

I am trying to show that the linear transformation of two monomially-equivalent linear codes preserves the minimum distance and the two equivalent codes have the same dimension. First, what is the ...
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80 views

How to find $(Ker(A^{*}))^{\perp}$

Let $$A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & -2 & -1 \\ 1 & 2 & 4 & 3 \end{pmatrix}$$ Find a basis for $(Ker(A^{*}))^{\perp}$. Find vectors $b_i$ such that $ ...
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53 views

Prove a linear transformation symmetric and positive

Consider the linear transformation of $\mathbb{R}^3$ given by $Ax = (a \cdot x)a+ |a|^2x$. Is $A$ symmetric? Is it positive? I know that a matrix is symmetric IFF $(Ax,y) = (x,Ay)$ and positive if ...
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42 views

Linear Alegbra - Is this linear transformation isomorphism?

Let $\mathbb R^4 \rightarrow \mathbb R^4$ linear transformation. That : $$\dim\operatorname{Im}(T+I)=\dim\ker(3I-T)=2$$ Is $T-I$ isomorphism? The only thing I come up with is that 3 and -1 are ...
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53 views

Changing the basis of a matrix of the linear mapping

Let $$A= \begin{pmatrix} 1 & 3 & 1 & 4 \\ 2 & 3 & 4 & 5 \end{pmatrix}$$ be the matrix of the linear mapping $F: \mathbb{R}^4 \to \mathbb{R}^2$ in the usual bases of ...
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36 views

Controllable and observable

The square matrices $A$ is invertible, $Q$ and $G$ symmetric positive semidefinite. Moreover, $(A,G)$ is controllable, and $(Q,A)$ is observable. I have the following question Is $(-A,-G)$ ...
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73 views

Im(A) = Im(A*A)

How does one prove that $Im(A^*) = Im(A^*A)$ and that the $Im(A) = Im(AA^*)$? I have found that $Ker(A) = Ker(A^*A)$ and that $Ker(A)=Ker(AA^{*})$. Also, the $Rank(A) = Rank(AA^{*}) = Rank(A^{*}A)$. ...
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106 views

Linear Alegbra - Find Base for ImT and KerT

Find basis for $ImT$ and basis for $kerT$. $v_1=(1,1,1)$ $v_2=(0,0,1)$ $v_3=(0,1,1)$ $B=(v_1,v_2,v_3) \in R^3$ My solution $[T]_B=\left[\begin{array}{cccc} 1 & 2 & 1 \\ -2 & 0 & ...
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20 views

If $\mu(e_1,…e_n)=1$, then how to show that $\mu=f^1\wedge f^2…\wedge f^n$?

Let V be a n dimensional vector space, $\mu$ be an antisymmetric n tensor.(i.e, a real valued multilinear functional with n inputs) If there exists a basis for $V$, say, {$e_1,e_2,...,e_n$}, such that ...
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Find the projection of any vector onto the linear span and the normal from any vector to that span

Show that the vectors $u_1 = (1/9,4/9,8/9), u_2=(8/9,-4/9,1/9), u_3=(-4/9,-7/9,4/9)$ form an orthonormal basis of $\mathbb{R}^3$. Find the projection of any vector $x=(\xi_1,\xi_2,\xi_3) \in ...
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36 views

Prove a condition from properties of matrices

Suppose that $A$ is an $n\times n$ real symmetric matrix of rank $n-1$. Suppose also that $Au=0$, where $u=(1,1,...,1)^T$. Show that $Ax=y$ has a solution if and only if $\sum_{j=1}^ny_j=0$. ...
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149 views

Show that $P\colon S^3\to SO(3)$ is a covering map.

Please help, anyone? This I have done so far: identify $S^3$ with the quaternions of unit length and identify $R^3$ with the pure quaternions, that is, those of the form $\{b_i, c_j, d_k\}$, for ...
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1answer
30 views

Linear Alegbra - Orthonormal Basis

Let $v_1=(1,2,2)$, $v_2=(1,0,1)$ in $R^3$ Find an orthonormal basis $\{u_1,u_2\}$ to $Sp\{v_1,v_2\}$ Find vector $w \in Sp\{v_1,v_2\}$ so $w \bullet u_1=3$ and $w \bullet u_2=9$ My solution I'm ...
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35 views

Linearize non linear function

Is it possible to linearize the function $f(x) = 1-exp(\frac{x}{b})$ so that one could use it in a linear regression?
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General Advice on the proof of inequalities

I've got a pretty basic question that has been slightly confusing to me (I kinda understand this, but I need some affirmation). Basically, say I have an inequality $(2n+3)^2>4(n+1)(n+2)$, which, ...
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Linear Algebra - $n\times n$ matrices [closed]

Let $A$ and $B$ be any $n \times n$ defined over the real numbers. Let assume that $I+AB$ invertible matrix. Prove : $I+BA$ invertible matrix $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ Any help will be ...
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1answer
69 views

Basis for the range of the matrix

I did the first part properly and showed that the rank is 2, but putting this matrix into a reduced row echleon form. For the second part, I get the wrong basis vectors simply because of ...
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54 views

How to expand inner product square?

How does this $||x-x'||$ expand to the equation below? $\|x-x'\|^2 = (x^T)x + (x')^T x' - 2x^T x'$
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45 views

Matrices and bases

Can you please verify my argument: Let $M = \begin{pmatrix} a & b\\ c& d\end{pmatrix}$, where $a,b,c,d$ are all real. $$AM=\begin{pmatrix} c & d\\ a& b\end{pmatrix}$$ Let $B$ be ...
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span{AB-BA }=set of all matrices of trace 0

This is a question from one of our class tests. let W be the space span{AB-BA} where A and B are square matrices . and let H be the space of all square matrices of trace 0 . then prove that W=H. the ...
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58 views

Rational solution to AX=0

Let $\mathcal{M}_{n,p}(\mathbb{K})$ be the set of matrices $n\times p$ with coefficients in $\mathbb{K}$. Let $A\in\mathcal{M}_{n,p}(\mathbb{Q})$. We suppose there exists a non zero solution ...
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1answer
52 views

Why are the spectral norm of $A^{*}A$, $AA^{*}$ and $A$ equal?

I'm learning matrix norm now, but i don't have learned Hermitian before. Is there any theorem about hermitian i can use to prove that three matrices norm are equal?? Thanks a lot.
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Consider $A_{n\times n}$ and $B_{n\times n}$. If $AB = I_n$, are the columns of $A$ or $B$ linearly independent?

So we have that $AB = I_n$. This means that $A = B^{-1}$ and $B = A^{-1}$. In particular, we know that an inverse of both $A$ and $B$ exists. Thus, $rank(A) = rank(B) = n$. We know that the $im(A)$ or ...
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1answer
36 views

Consider nonzero $A_{2\times 2}$ such that $A^2 = \vec{0}$. Prove or disprove that dim$($ker$(A)) = 2$

I'm having a couple of difficulties with this problem. The first comes with finding an $A$ such that $A^2 = \vec{0}$. In the case of a $2 \times 2 $ matrix, it seems simple enough to try a few ...
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37 views

Help identifying a theorem about representations of an orthogonal linear transformation

I'm looking for some information on a theorem presented in my linear algebra notes stated without name - basically it states that (taken verbatim): If $f: V \rightarrow V $ is an orthogonal linear ...
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27 views

$\|a\|\le 1\Leftrightarrow -I\le a\le I$, where $a$ is a hermitian matrix

need to prove the last statement, I know $\|a\|=\max\{|\lambda|: \lambda \text{ is an eigen value of } a\}$ suppose $-I\le a\le I$, then as $a$ is hermitian there exists unitary matrix $u$ and a ...
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148 views

Give conditions on a,b,c, and d such that A has two, one, and no eigenvalues?

I am given that matrix $$A= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ and I need to find conditions on a,b,c, and d such that A has Two distinct ...
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3answers
103 views

Find the eigenvalues of the matrix and give the bases for each of the corresponding eigenspaces

I'm having issues with this problem. I have solved for the eigenvalues but am having trouble finding the bases for both eigenvalues. The pictures below contain my work for solving for the eigenvalues ...
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1answer
831 views

Geometric Interpretation of Eigenvectors

I just want to make sure I'm thinking about this correctly. I've been given a matrix A and I need to find the eigenvalues and eigenvectors geometrically. I have found the eigenvalues. It wasn't too ...
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1answer
23 views

Working with vectors in Linear Algebra

I'm just kinda confused about a problem in my linear algebra textbook. Maybe one of you geniuses on here can help me out. ...
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57 views

Show that $V$ is a vector space

If we let $$V = \{ x \mid x = \begin{bmatrix} x_1 \\[0.3em] x_2 \end{bmatrix},\text{ where }x_2 > 0 \} $$ and define addition and scalar multiplication by $$u + v = ...
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1answer
28 views

How to find dimension of a subspace?

For instance, take $S=\{\mathbf{v}\in\mathbb{R}^5:\;v_1+v_2+v_3=0,\;v_1+v_2+v_5=0\}\subset\mathbb{R}^5.$ How would I go about finding $\dim S?$ I can see that both $v_1+v_2+v_3=0,\;v_1+v_2+v_5=0$ ...
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1answer
37 views

We can find at most $x$ linearly independent vectors that span a subspace $V$

Consider vectors $\vec{v_1},\ldots,\vec{v_p}$ and $\vec{w_1},\ldots,\vec{w_q}$ in a subspace $V$ of $\mathbb{R}^n$. If the vectors $\vec{v_1},\ldots,\vec{v_p}$ are linearly independent, and the ...
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1answer
132 views

Whats wrong with this proof of Cayley-Hamilton

There are a handful of longish proofs for Cayley-Hamilton, but I haven't seen one that goes along the following lines. What's wrong in my thinking. Take a real square matrix $A$, with Jordan ...
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62 views

Linear Algebra question relating to eigenvectors

Let A be an m x m positive definite symmetric matrix with eigenvalue-eigenvector pairs $(\lambda_1,e_1),....,(\lambda_m,e_m).$ The eigenvectors are orthonormal. Let $C = e_1e_1'+....+e_me_m'$. ...
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1answer
22 views

If S and T are transformattion mappings, what is [ST]?

S and T are transformation mappings, what does [ST] and [TS] mean? Does it mean transform via S and then apply T to the result and vice versa?
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101 views

Understanding how to solve a Cost Function?

I'm having trouble seeing the relationship in the following equation. Let's assume $J(0,1)$ and $m=4$. First I figure out my hypothesis function ...
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1answer
61 views

Can you divide by matrix norms in an equation?

Supposing that a matrix A has an eigenvalue lambda, show that for any induced matrix norm, $||A|| \geq |\lambda|$. I attempted the solution, but I am not sure if it is valid to cancel the norm of ...
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1answer
37 views

Linear Algebra - Invertible matrice

I have this problem and I'm not sure my solution is correct. Let $A$ be any $n \times n$ matrix, defined over the real numbers, A is not invertible matrix. Proof that there's is $B \neq 0$ and $C ...
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How to Determine which subsets of R^3 is a subspace of R^3.

I have some questions about determining which subset is a subspace of R^3. Here are the questions: ...
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53 views

Tensor analog of Matrix Product

Given two $n \times n$ matrices $A$ and $B$, we can form their matrix product in the usual way. Is there a similar product for tensors? E.g., if one is given two $n \times n \times n$ tensors ...
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444 views

Proof that a common brain teaser is wrong (Burning Rope)

There is a common brain teaser that goes like this: You are given two ropes and a lighter. This is the only equipment you can use. You are told that each of the two ropes has the following property: ...
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Why isn't the quotient space $V/V = \{ V \}$?

If $W \subset V$, then one defines the quotient space, $$V/W = \{ v + W : v \in V \}$$ So why isn't this right? $$V/V = \{v + V : v \in V \} = \{V \}$$? I read that $V/V = \{ 0 \}$? Why can't the ...
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Linear Algebra - Invertible matrices and determinants

Let $A$ be any $n \times n$ invertible matrix, defined over the integer numbers. Let assume that $A^{-1}$ (Inverse of A) is also defined over the integer numbers. Prove that $\det A\in\{-1,+1\}$. ...
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24 views

Switching Sides in Expression

Please forgive me if my terminology is incorrect. What particular rule is used for the following: (6a - 10c) to -(10c - 6a) I have looked over a few textbooks but I am not sure why the above is ...
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45 views

Creating a system of linear equations.

An average mark is computed for 100 students in Business, an average is computed for 300 students in Arts, and an average is computed for 200 students in Science. The average of these three averages ...
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49 views

Is every matrix in $F[x]^{n\times n}$ is row-equivalent to an upper-triangular matrix?

I'm trying to solve this problem, but I don't have any idea of what I should do.. True or false? Every matrix in $F[x]^{n\times n}$ is row-equivalent to an upper-triangular matrix. This is a ...
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102 views

How do I solve a multivariable equation?

How could I solve for variables $x$, $y$, $z$, and $w$ for the equation $$ax+by+cz+dw$$ With given values $a$, $b$, $c$, and $d$. For example, how would I find a set of potential values for $x$, ...
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1answer
61 views

When does a square matrix have an eigen-decomposition? When is a matrix defective? [duplicate]

Some square matrices, like $ \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right)$, don't have a complete set of eigenvectors. By complete I mean that the eigenvectors span the entire ...