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Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

Linear algebra is concerned with vector spaces and linear transformations between them:

$$(x_1, \dots, x_n)\to a_1x_n+\dots+a_nx_n$$

Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization and Jordan normal forms.

This is a general tag, most of the subjects included have secondary tags, e.g.

127034 questions
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Finding the additive inverse in a vector space with unusual operations

Let $V=\mathbb{R}$. For $u,v\in V$ and $a\in\mathbb{R}$, define vector addition by $$u\boxplus v:=u+v+2$$ and scalar multiplication by $$a\boxdot u:=au+2a−2.$$ It can be shown that $(V,\boxplus,\boxdot)$ is a vector space over the scalar…
KnowledgeGeek
  • 327
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3 answers

Proving any linear map on a subspace of $V$ can be extended to a linear map on $V$

Suppose V is a finite dimensional vector space. I am trying to prove that any linear map on subspace of V can be extended to linear map on V. Basically, showing that if $U$ is a subspace of $V$ and $S \in L(U,W)$, then there exists a $T \in L(V,W)$…
user123276
  • 3,445
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What properties should a matrix have if all its eigenvalues are real?

Recently, I’m trying to prove all the eigenvalues of a class of matrices are real. The matrices are complex and not hermitian. The problem for me is I don't know any properties for a matrix with all the eigenvalues real. So would you please tell me…
user1992738
  • 63
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Show that $0$ and $1$ are the only real eigenvalues of $A.$

Let $A\in M_n(\mathbb R)$ such that $A^2=A^T.$ Show that $0$ and $1$ are the only real eigenvalues of $A.$ All I can see is that $\det A=0$ or $1.$ I can't proceed any further.
Shantipriya
  • 107
6
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How is b not a linear combination of these vectors?

Determine if $\vec b$ is a linear combination of $\vec a_1,\vec a_2,\vec a_3$. $\vec a_1 = \left[\begin{array}{c} 1 \\ -2 \\ 0 \\ \end{array}\right], \vec a_2 = \left[\begin{array}{c} 0 \\ 1 \\ 2 \\ \end{array}\right], \vec…
Bobby Lee
  • 1,030
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1 answer

Linear Independence easy question

I have these vectors $B = \{u, v, w\}$ with $$u = (-1, 1, -1),\, v = (19, 10, -9),\, w = (-1, x, y)$$ And i want to prove that these vectors are linearly independent. I have no problem to proove that three vectors without unknown variables are…
Manos
  • 185
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How to compute the similarity transformation matrix

Stuck on this question: Let $$A=\begin{pmatrix} 2&1\\ -1&-1 \end{pmatrix}$$$$B=\begin{pmatrix} -2&5\\ -1&3 \end{pmatrix}$$$$C=\begin{pmatrix} 5&2\\ 4&1 \end{pmatrix}$$ Show that A is similar to B, but that A is not similar to C. I can do the…
George1811
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What conditions must the constants b1,b2 and b3 satisfy so that the system below has a solution

$$x_1 + 2x_2 + 3x_3 = b_1 \\ 2x_1 + 5x_2 + 3x_3 = b_2 \\ x_2 - 3x_3 = b_3$$ Use Gauss method: $-2p_1+p_2$ to produce $x_2-x_3=-2b_1+b_2$. $p_1+p_3$ to produce $x_1+3x_2=b_1+b_3$ $-p_2+p_1$ to produce $-x_1-3x_2= b_1-b_2$ The above produces the…
user983246
  • 617
6
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2 answers

Skew-symmetric matrix and quadratic form

Show that the quadratic form of a matrix is $0$ if and only if the matrix is skew– symmetric, i.e., show that $q_A(x) = 0$ for all $x$ iff $A^t = −A$. Thanks a lot!
BVFanZ
  • 839
6
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2 answers

Example of a non-linear isometry?

Is there a simple example of an isometry between normed vector spaces that is not an affine map?
Name
  • 61
6
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2 answers

Bijective check with matrix

My book doesn't cover the criterion for bijective transformations very well. I just want to check my understanding: is this statement true? Let F be a linear transformation. Let A be the matrix that represents that transformation (which means that…
jacob
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How to show that you can find two subspaces that don't intersect

Suppose $V, V'$ are subspaces of dimension $d$ of a vector space $X$. Then there is a subspace $W$ of $X$ of codimension $d$ such that $W \cap V = W \cap V' = { 0 }$. This can be proved by choosing an explicit basis for $X$ which contains a basis…
Akhil Mathew
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Does there exist an infinite $S \subset \Bbb R^3$ such that any three vectors in $S$ are linearly independent?

Does there exist an infinite subset $S \subset \Bbb R^3$ such that any three vectors in $S$ are linearly independent?
Struggler
  • 2,554
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3 answers

In a finite dimensional vector space a positive operator can have infinitely many square roots

"In a finite dimensional vector space a identity operator (which is positive) can have infinitely many square roots". I see only $2 ^ {C(n,2)}$ when dimension is $n (>1)$ by swapping two basis elements. Where am I wrong ?
user103623
  • 558
6
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1 answer

Column Space of AA' is equal to column of A.

This is my question. How to show that the column space of matrix A is just equal to the column space of AA'?.. A' represents the transpose of A. I know that the column space of AA' is a subset of the column space of A which is just trivial. But the…
Aintegral
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