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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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Find the image of a vector by using the standard matrix (for the linear transformation T)

Was wondering if anyone can help out with the following problem: Use the standard matrix for the linear transformation $T$ to find the image of the vector $\mathbf{v}$, where $$T(x,y) = (x+y,x-y, 2x,2y),\qquad \mathbf{v}=(3,-3).$$ I found out…
user7814
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5 answers

Are strictly upper triangular matrices nilpotent?

An $n\times n$ matrix $A$ is called nilpotent if $A^m = 0$ for some $m\ge1$. Show that every triangular matrix with zeros on the main diagonal is nilpotent.
Bowen
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Given a reduced row exhelon form of a $4 \times 4$ matrix and two columns, how do you find the other two columns?

I am given the following : Let $A$ be a $4 \times 4$ matrix with RREF given by: $$ U = \begin{bmatrix} 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}. $$ If $$ a_1 = \begin{bmatrix} -3 \\ 5 \\ 2 \\ 1…
rphello101
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The dimension of the sum of subspaces $(U_1,\ldots,U_n)$

If $U_1$ and $U_2$ are subspaces of a finite dimensional vector space, then $$\dim(U_1+U_2) = \dim U_1+\dim U_2-\dim(U_1 \cap U_2).$$ How can one generalize this notion to a collection of $n$ subspaces $U_1,\ldots,U_n$? Or what does…
St Vincent
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There is a subspace $W$ of $V$ such that $V = U \oplus W$

I have a question about the following proof of the statement that for every subspace $U$ of a finite dimensional vector space $V$ there is a subspace $W$ of $V$ such that $V = U \oplus W$. Proof: Because $V$ is finite-dimensional, so is $U$. Thus…
St Vincent
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Uniqueness of LDU Factorisation [Strang P105 2.6.18]

Let $L$ be a lower triangular matrix, $D$ diagonal, and $U$ upper triangular. If $A = LDU$ and also $A = L_1D_1U_1$ with all factors invertible, then $L = L_1$ and $D = D_1$ and $U = U_1$. 'The three factors are unique!' Hint: Derive…
user53259
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1 answer

If $A^2=-I$, Prove that $\det{A}=1$

If $A^2=-I$ , where $A$ is a square matrix of order $n$ and which contains real entries only and $I$ is identity matrix. Then how can we prove that $\det(A)=1$?. I could prove that $n$ should be an even integer. But could not proceed to prove that…
thehe
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Proof verification

I have a problem and a proposed solution. I want to know if I have done it correctly. Problem Statement: Let $V=F^n$ be the space of column vectors. Prove that every subspace $W$ of $V$ is the space of solutions of some system of homogeneous linear…
user85362
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2 answers

Linear Algebra: finding a basis for a subspace of $\mathbb{R}^4$

So I am stuck on this example from my Into. Linear Algerbra book I'm not exactly sure how I'm supposed to find the basis in this case. Am I just supposed to use a random t and s value and call the single vector a basis? (There were no previous…
user7814
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2 answers

Geometry of linear equations

its my first question here. I'm re-self-studying linear algebra from different sources and one of them is Linear Algebra and Its applications by g.strang 4th ed. . While i have studied a bunch of material i still don't grasp some basics and i really…
MatVe
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Determine whether there's permutation $p$ such that for matrices $A_{i}$, $A_{p_{1}}A_{p_{2}} \dots A_{p_{n}} = B$

Given $m$ $n\times n$ matrices $A_{1},A_{2} \dots A_{m}$ and a matrix $B$, is there a way to determine whether there's permutation $p$ such that for matrices $A_{i}$, $A_{p_{1}}A_{p_{2}} \dots A_{p_{m}} = B$. The solution should be "almost correct"…
rddccd
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$f$ and $h\circ f$ are linear and $f$ is surjective, is $h$ linear?

I have a very easy question but I can't find the solution. Let $V,U,W$ be three $\mathbb{R}$-vector spaces and let $f: V \rightarrow U$ be a surjective linear map and $g: V \rightarrow W$ a linear map. Now, define $h : U \rightarrow W$ such that $h…
eomp
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4 answers

Minimal polynomial

Let $V$ be the vector space of square matrices of order $n$ over the field $F$. Let $A$ be a fixed square matrix of $n$ and let $T$ be a linear operator on $V$ such that $T(B) = AB$. Show that the minimal polynomial for $T$ is the minimal…
prasad
  • 517
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$AB$ is not invertible

Is it true that if $A$ is an $m \times n$ matrix and $B$ is an $n \times m$ matrix, with $m > n $ then $\det(AB)=0$?
Twnk
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4 answers

Could a set of $3$ vectors in $\mathbb{R}^4$ span all of $\mathbb{R}^4$?

Could a set of $3$ vectors in $\mathbb{R}^4$ span all of $\mathbb{R}^4$; is this the same as asking if a 4 x 3 matrix could span $\mathbb{R}^4$ or if a 3 x 4 matrix could span $\mathbb{R}^4$?
spitfiredd
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