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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Define: A solution of a linear equations system + Row, Column & Null spaces relations

The linear equations system: $$\left(\begin{array}{ccc|c}1 & 1 & 1 & 3 \\1 & 2 & 3 & 6 \\1 & 3 & 5 & 9\end{array}\right).$$ Has the following solution: $$ ...
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57 views

The set of differentiable functions such that $f'(2)=b$ is a linear subspace if and only if $b=0$

Questions are in bold. The set of differentiable real-valued functions on (0,3) such that $f'(2)=b$ is a subspace of $(0,3)\to \mathbb R$ if and only if $b=0$ ($(0,3)\to \mathbb R$ denotes the set ...
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1answer
13 views

Linear Transformations on Infinite Dimensional Vector Spaces

Let $T$ be a linear transformation $T:V\to V$, where $V$ is an infinite dimensional vector space. How can we construct examples such as $1.$ T is one to one but not onto $2.$ T is onto but not ...
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U=span{(1,1,-1), (2,3,-1), (3,1,-5)} and V=span{(1,1,-3), (3,-2,-8), (2,1,-3)}. What is U $\cap$ V

Let U=span{(1,1,-1), (2,3,-1), (3,1,-5)} and V=span{(1,1,-3), (3,-2,-8), (2,1,-3)}. What is U $\cap$ V? 1. U 2. V 3.zero subspace 4. None of these. I tried firstly to find dim of U $ \cap$ V , by ...
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31 views

Reversed Cayley transformation for any unitary matrix

It is well known that if $Q$ is a complex unitary matrix such that $I+Q$ is invertible (where $I$ is the identity matrix), that is, $-1$ is not an eigenvalue of $Q$, then $$ A:=(I-Q)(I+Q)^{-1} $$ is ...
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Diagonalization of a quadratic form with parameter $k \in \mathbb{R}$: $q(x,y,z)=(2+k)x^2+2y^2+kz^2+4xy-2kxz$

Let $q: \mathbb{R^3} \to \mathbb{R}$ be the quadratic form $$q(x,y,z)=(2+k)x^2+2y^2+kz^2+4xy-2kxz,$$ with $k \in \mathbb{R}$. I would like to diagonalize this form and then write it in the canonical ...
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9 views

Linear least-squares with matrices rather than vectors

I have two coordinate frames, each represented by a 4-by-4 matrix ($A$ and $B$), where this is the pose (orientation and translation) in homogeneous coordinates. I now want to find a third matrix $T$, ...
3
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3answers
42 views

Rank in row echelon form

$$A= \begin{bmatrix} a & 1 & a & 0 & 0 & 0 \\ 0 & b & 1 & b & 0 & 0 \\ 0 & 0 & c & 1 & c & 0 \\ 0 & 0 & 0 & d & 1 ...
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1answer
39 views

Find the value of the Determinant

If $a^2+b^2+c^2+ab+bc+ca \le 0\quad \forall a, b, c\in\mathbb{R}$, then find the value of the determinant $$ \begin{vmatrix} (a+b+2)^2 & a^2+b^2 & 1 \\ 1 & (b+c+2)^2 ...
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Matrix notation of vectors?

My linear algebra book says that a vector is a one-column matrix. However, in the context of what we are studying (linear equations) it would make more sense if a vector was of the form of the ...
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1answer
16 views

Kernel and image of a diagonalizable endomorphism $f$ given only an orthogonal basis$B={w_1,w_2,w_3}$, an eigenvalue, and that $f(w_1)=f(w_2)$

Let $f: \mathbb{R^3} \to \mathbb{R^3}$ be diagonalizable with $B$, basis of eigenvectors, such that $B={w_1,w_2,w_3}$, where $w_1=(1,2,0),w_2=(0,1,1),w_3=(0,1,-1)$. If we know that $3$ is an ...
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2answers
68 views

A linear operator $T: V \rightarrow V$ commuting with all linear operators is a scalar multiple of the identity. [duplicate]

Let $\mathbb{K}$ a field, $V$ a vector space over $\mathbb{K}$. If $T:V\to V$ commutes with all other linear operators $V \to V$, then there exists $\lambda \in \mathbb{K}$ such that $T= \lambda I$, ...
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1answer
17 views

Proving that a matrix product is singular

I just played around in mathematica and found out that it seems like if $A$ is an $m\times n$ matrix and B is an $n\times m$ matrix, with $m>n$, then $AB$ is singular. How does one go about proving ...
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7answers
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why don't we define vector multiplication component-wise?

I was just wondering why we don't ever define multiplication of vectors as individual component multiplication. That is, why doesn't anybody ever define $\langle a_1,b_1 \rangle \cdot \langle a_2,b_2 ...
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2answers
41 views

Matrices that commute with all matrices [duplicate]

Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n $ matrices. Show that $$Z_n = \{\lambda I_n \ | \ \lambda \in \mathbb R\}$$ ($I_n$ is the $n \times n$ identity ...
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0answers
7 views

Decomposition of 4x4 or larger affine transformation matrix to individual variables per degree of freedom.

There are a couple of problems and solutions where affine matrices are decomposed into their seperate tranformations. However they are all for the 2D case and I`m finding it difficult to generalise it ...
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1answer
17 views

Linear Transformations: Proving 1 dimensional subspace goes to 1 dimensional

I am having trouble understanding this whole question, and how to prove it. Let $F:\mathbb{R}^n\to\mathbb{R}^m$ be a linear transformation. Prove that if $L$ is a $1$-dimensional subspace of ...
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2answers
55 views

Solving a system of three simultaneous equations

Given the system $$ \begin{align*} -2x + ay - bz &= -4 \\ x + bz &= 2 \\ 2x + y + 3bz &= b \end{align*} $$ The question asks to find conditions on $a$ and $b$ that the system has no ...
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1answer
28 views

Property of orthogonal and skew symmetric matrix

If $A$ be a $n\times n$ orthogonal matrix and $X$ be a matrix such that $X=(A+I)^{-1}(A-I)$ then show that $X$ is a skew-symmetric matrix,whenever $n$ is an odd integer.
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$Ker(T) \subseteq Ker(S)$ implies the exist some $H$ s.t $H\circ T=S$

Let $V,W$ be a vector space over $\mathbb{F}$, with finite dimension. Let $T,S:V\rightarrow W$ linear transformations such that $Ker(T)\subseteq Ker(S)$. Prove that exists some linear transformation ...
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how to conclude a subset of $M_n(\mathbb{C})$ is compact from spectral radius?

could any one tell me which of the following is/are compact subset? $S=\{A\in M_n(\mathbb{C}): \rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): A=A^*,\rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): ...
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1answer
31 views

set of all $2\times 2$ matrcies having neither eigen value is real

Could any one tell me whether the following subsets of $M_2(\mathbb{R})$ are open, closed or neither open nor closed? set of all $2\times 2$ matrcies having neither eigen value is real. set of all ...
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2answers
21 views

Interchange rows in a matrix without using interchange operation

I'm sure that it's already out there somewhere in the abyss that is page 37 on google, so I apologize. I haven't been able to find it. Given some arbitrary matrix, how can two rows be interchanged ...
3
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2answers
433 views

Equation of third side of Triangle

A Triangle is formed by Pair of lines $$ ax^2+2hxy+by^2=0$$ and a third side L3.Given the Orthocentre of Triangle is $$(c,d)$$, Find Equation of Third Side.
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21 views

Given a bilinear (or quadratic) form, how can you find the orthogonal of a vector space?

Let $V$ be a vector space over a field $F$ equipped with a symmetric bilinear form $B$. Let $W$ be a vector subspace of $V$. I know that we define the orthogonal complement $W^\bot$ to be ...
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2answers
24 views

Dimension of $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$?

What is the dimension of an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$? Do I have to make distinct cases with as $p + q < n$ and equal to $n$? And if their ...
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1answer
11 views

Conditions of invertibility, linear transformations

Please, I need a hint. :) Let $T:\Bbb R^m\rightarrow \Bbb R^n$ and $ U:\Bbb R^n \rightarrow \Bbb R^m $ be linear transformations. What are the conditions that $m, n$ have to satisfy to $UT:\Bbb R^m ...
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1answer
42 views

Finding a basis for Kernel and Image of a linear transformation using Gaussian elimination.

This is a very fast method for computing the kernel/nullspace and image/column space of a matrix. I learned this from my linear algebra teacher but I haven't seen it mentioned online apart from this ...
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0answers
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What's the degree of freedom of this kind of matrix?

We first have a unitary matrix in $\mathbb{C}$ $$U = \{a_{ij}\}_{n\times n},$$ where "unitary" means $$U'U = I, \quad U'\text{means conjugate transpose.} $$ I know how to calculate its degree of ...
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3answers
48 views

I have difficulty understanding functions forming vector space.

I have knowledge of basic linear algebra, so I can understand the finite vector space as linear combinations of vectors of $R^n$. However, when it comes function as vector and functions form a ...
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1answer
26 views

Find basis of subspaces

I don't know how to create basis of V1 and V2. If I want to prove M1^2=M1, do I need to find matrix representation of M1 first? Thanks!!!!!!
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1answer
22 views

How to prove that the infinity norm of a matrix is the max of row sum?

I know how to prove that the 1-norm of a matrix is the max of the column sum, but not sure how to prove that the inf-norm is the max of the row sum. Any suggestion? Thanks
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2answers
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Linear Transformation one to one and onto?

Let $A = \left[ \begin{array}{ccc} 5 & -4 & 5 \\ 1 & -2 & -1 \\ -1 & 5 & 6 \end{array} \right].$ Is the linear transformation $T : \mathbb{R}^3 → \mathbb{R}^3$ defined by ...
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1answer
30 views

Reflexive bilinear forms.

Let $V$ be a vector space and $B: V \times V \to \Bbb R$ be a bilinear form. Usually, I see books defining that if $B$ is symmetric, vectors ${\bf u},{\bf v} \in V$ are $B$-orthogonal if $B({\bf ...
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1answer
24 views

Definition of complex conjugate in complex vector space

I am starting reading about Hodge theory and while reading the definition of abstract Hodge structure a very basic question came to my mind... What is the definition of the conjugate of a subspace of ...
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2answers
23 views

How to “compare” vectors?

I'm reading the definition of matrix norm in Golub & Van Loan and came across this "It is clear that the p-norm of matrix A is the p-norm of the largest vector obtained by applying A to a unit ...
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2answers
36 views

Four Fundamental Spaces Question

Why do the four fundamental sub spaces come from Ax=b and A'y=f? Why these two equations?
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1answer
20 views

Intersection of “positive” open half-spaces

Prove that the intersection of "positive" open half-spaces associated with any basis $x_1,x_2, \ldots, x_n$ of a finite dimensional vector space $V$ is non-empty. Recall that the "positive" open ...
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3answers
122 views

Complex square matrices. Difficult proof.

$det(I+A\cdot\bar{A}) \ge 0$ Is it possible to prove the inequality is true for all complex square matrices $A$ where $I$ is the identity matrix and $\bar{A}$ is the complex conjugated matrix.
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If two linear functionals are such that the kernel of one is contained in the kernel of the other, then they are proportional [duplicate]

Let $V$ be a vector space over $K$ and let $f,g \in V^*$ and satisfy $\ker f \subseteq \ker g$. Show there exist such $c \in K$ so that $c \cdot f =g$ How to approach this problem ?
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consider if given vectors are elements of the span?

Consider the vectors u = (1,3,2) and v = (2,-1,1) in ℝ³. Determine whether or not (1,7,5) ∈ span(u,v) . Not really sure what to do, I was thinking of checking to see if u and v span ...
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1answer
24 views

Confused about Fourier series?

From linear algebra we know that if a set of vectors form a basis for a space, their is a unique linear combination of the basis to form any vector in that space. I'm assuming this extends to scalar ...
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3answers
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Show that a linear map $f:A\to B$ such that $\mathrm{dim}\,A> \mathrm{dim}\,B$ can't be 1-1. [on hold]

Given a linear map $f:A\to B$ such that $A, B$ are vector spaces and $\mathrm{dim}\,A> \mathrm{dim}\,B$, show that $ \ f$ can't be 1-1.
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1answer
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Kernel of $q(x,y,z)=2x^2-4xy+2z^2-4xz+4yz$

I have some problems when calculating the kernel of the quadratic form $q(x,y,z)=2x^2-4xy+2z^2-4xz+4yz$: indeed, I get $Ker=\{(x,y,z)|x^2-y^2=0\}$, which results in a 2-dimensional kernel. Could you ...
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0answers
22 views

How to solve the vector differential equation? [on hold]

I'm new to this section, so I'm trying to solve vector differential equations, and I need some guidance. Could anybody give a step-by-step process for doing so, so that I could do some more problems ...
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1answer
19 views

Matrix representation induced by quotient space

someone can help me with this question, I know how to solve ker(A) but I don't know how to develop matrix representation. Thanks!!!!!
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2answers
27 views

Sense of rotation. How would the rotation matrix look like for this “arbitrary” axis?

My first question is how do you define the sense of rotation about an arbitrary axis? Rotations are usually counterclockwise and when referring to rotation with respect to the $x$,$y$ or $z$ axis ...
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1answer
36 views

What is the derivative of (Ax)'

Let $f(x)=(Ax)^T$ where A is a matrix and x is a vector. How do you explain that $f'(x)=(Ax)^T$? Specifically, that $\frac{\partial}{\partial x} f(x) (y) = (Ay)^T$. I can't seem to do it rigorously. ...
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3answers
18 views

Find all linear operators such that $F^2 = F$ and $F(x,y) = (ax,bx+cy)$

I need to find all linear operators that match $F^2 = F$ and $F(x,y) = (ax,bx+cy)$ *where $F^2$ means $F$ composed with itself. So what I did: $F(x,y) = (ax,bx+cy)\implies F(F(x,y)) = ...
7
votes
4answers
792 views

Show that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$

I know about the fact that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$, where $A$ and $B$ are $m \times n$ matrices. But somehow, I don't find this as intuitive as ...