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 ...

learn more… | top users | synonyms

0
votes
0answers
10 views

Linear algebra homogenous system

Given a $3\times3$ matrix depending on a real parameter $x$. Denote by $S(A(x))$ the space of all solutions of the homogenous system $A(x)=0$. How can one find this space in general?
0
votes
1answer
10 views

Respresting linear transformation with matrix with restrictions

When given a set of restrictions, what is the way to find a representing matrix of a linear transformation? Lets say I have T:R^4->R^3 and I need the Ker(T) to be spaned by {(1,2,3,4), (0,1,1,1)}. ...
2
votes
0answers
14 views

When does $\| \Pi \|_1 = 1$ where $\Pi$ is a projection.

By projection I mean any matrix such that $\Pi = \Pi^2$. It is well known that all projections can be written as $\Pi = A(B^\top A)^{-1}B^\top$ for some $A,B$. Characterize the class of projections ...
1
vote
2answers
16 views

Show that if $v\in (V_c)^{\perp}$ then $(Av)\in (V_c)^{\perp}$ for a normal matrix $A$ with an eigenvalue $c$

Suppose $A \in M_{n\times n}(\mathbb C)$ is a normal matrix and $c$ is an eigenvalue of $A$. I'm trying to show that if $v\in (V_c)^{\perp}$ then $(Av)\in (V_c)^{\perp}$. I know that if we were ...
0
votes
1answer
21 views

Misunderstanding in the proof that the sum of subspaces is the smallest containing subspace.

So if $V_1,...,V_n$ are subspaces of $M$ then $V_1+...+V_n$ is the smallest subspace of $M$ containing $V_1,...,V_n$ The proof is that clearly $V_1,...,V_n$ are all contained in $V_1+...+V_n$ Then ...
0
votes
0answers
18 views

Matrix representation of shape operator

Let $f$ be a parametrized surface $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and $N : \Omega \rightarrow Tf$ the Gauß map. Then the shape operator is defined as $L = -DN \circ Df^{-1}.$ ...
3
votes
2answers
97 views

How to show that $A^2=AB+BA$ implies $\det(AB-BA)=0$ for $3\times3$ matrices?

Let $A$ and $B$ be two $3\times 3$ matrices with complex entries,such that $A^2=AB+BA$. Prove that $\det(AB-BA)=0$. (Is the above result true for matrices with real entries?)
-1
votes
1answer
10 views

Finding base B'

I have B = {(0,2,1),(-2,2,1),(-1,2,1)} how can I find B' so $ x + [x]_B + [x]_{B'} = 0 $ (equlas zero vector). For every vector $ x \in \mathbb{R}^{3} $.
1
vote
2answers
30 views

Multiplication of rational with irrational number?

If $a$ is rational and $b$ is irrational number and we know that $ab$ is rational, then what can we say about $a/b$? Is true that it's equal to 0?
0
votes
1answer
22 views

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: $$ ...
0
votes
0answers
9 views

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 ...
0
votes
2answers
24 views

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 ...
1
vote
1answer
19 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 $T$, ...
1
vote
1answer
20 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 ...
2
votes
1answer
45 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 ...
3
votes
1answer
39 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 ...
0
votes
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 ...
1
vote
1answer
70 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 of ...
3
votes
3answers
46 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 ...
0
votes
0answers
9 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 ...
0
votes
2answers
42 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 ...
1
vote
2answers
98 views

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 ...
1
vote
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 ...
0
votes
0answers
19 views

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}): ...
2
votes
1answer
48 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 ...
0
votes
3answers
23 views

$Ker(T) \subseteq Ker(S)$ 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 ...
0
votes
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 ...
0
votes
1answer
29 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.
0
votes
1answer
17 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 ...
2
votes
0answers
26 views

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 ...
0
votes
2answers
27 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 ...
0
votes
3answers
49 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 ...
0
votes
1answer
31 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 ...
0
votes
2answers
25 views

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 ...
0
votes
1answer
24 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
3
votes
1answer
25 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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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!!!!!!
1
vote
0answers
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 ...
0
votes
2answers
15 views

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 ...
-1
votes
0answers
23 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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
1answer
23 views

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 ...
-2
votes
3answers
24 views

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.
0
votes
1answer
44 views

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 ?
1
vote
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!!!!!
1
vote
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)) = ...
0
votes
1answer
17 views

Kernel of the quadratic form $q(x,y,z)=3x^2+6xy+10xz+2yz+3z^2$

Let $q:\mathbb{R^3} \to \mathbb{R}$ such that $$q(x,y,z)=3x^2+6xy+10xz+2yz+3z^2.$$ I have to determine rank, signature, kernel, and the canonic form of q (with its matrix). I have solved most of the ...