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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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A question about linear operators

This is a problem from Hoffman and Kunze (Sec 3.4, problem 10, page 96). If T is a linear operator on $R^2$ defined by $T(x_1,x_2)=(x_1,0)$ we see that it is represented in the standard basis by the matrix A such that $A(1,1)=1$ and rest of the…
Shibi Vasudevan
  • 1,603
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How do I express vectors as linear combinations?

Basically, I'm stuck at this question for some time, and it goes like this: A, B, C, and D are consecutive points of a parallelogram. Point E divides the diagonal AC do that |AE|:|EC| = 1 : 3. Point F divides the diagonal BD so that |BD|:|BF| = 4 :…
KingKendrick
  • 41
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Looking for a basis of a vector subspace

Let $n \geq 4$ and $\mathbb{R_n}$[x] a vector space of all polynomials of degree n at most. There is given a set: $ U = \{p \in \mathbb{R_n}[x]; p(1) = p(-1), p''(0) = 2p(1)\} $ Find some basis of a vector subspace U and deterime $dimU$ My…
m259
  • 139
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Find a basis for the range of linear operator and linear transformation.

A) Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be the linear operator given by the formula $T(x,y) = (2x-y, -8x+4y)$. Find a basis for the range of the linear operator. B) Let $T: \mathbb{R}^4 \to \mathbb{R}^3$ be the linear transformation given by…
briteId
  • 353
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Equal null space of two linear operators implies existence of an invertible operator

The full question goes as follows: "Suppose W is finite-dimensional and T1, T2 $\in$ L(V,W). Prove that null(T1) = null(T2) if and only if there exists an invertible operator S $\in$ L(W) s.t. T1 = ST2. I know that there already have been questions…
Chapel
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Necessary and sufficient condition for Nilpotent Matrix

Let $\alpha$ and $\beta$ be nonzero elements in $\mathbb{F}(n,1)$. Then $A = \alpha\beta^T \in \mathbb{F}(n,n)$ Prove that a necessary and sufficient condition for $A$ to be nilpotent is $\alpha^T\beta=0$. If so, what is the degree of $A$.…
AAP
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Complex numbers linearly dependent over $\mathbb{R}$.

Sorry if this is really basic but for some reason it's giving me trouble. If $z,w,t \in \mathbb{C}$, do there exist real numbers $a,b,c \in \mathbb{R},$ not all zero, such that $$ az+bw+ct = 0 $$ I would imagine the answer is no generally because…
Irving Rabin
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Column Space of A

Suppose we have the matrix $A = \begin{bmatrix} 4 &6 \\ -8 &-12\\ \end{bmatrix} $ and the vector $b = \begin{bmatrix} 3\\ -6 \end{bmatrix} $. So I am tying to determine whether b lies in the column space of A. My answer for this problem is yes…
Jlhglkj Glkjhg
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Is there any proof of surjectiveness of matrix exponential using a density argument?

i showed that $$\begin{array}{ccccc} f & : & \mathcal{M}_n(\mathbb{C}) & \to & \mathbb{GL}_n(\mathbb{C}) \\ & & A & \mapsto & \exp(A)=\sum\limits_{k=0}^{\infty} \frac{A^k}{k!} \\ \end{array}$$ is a surjective application with the Jordan blocks. But…
vad koslov
  • 91
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Prove that A is idempotent without using Jordan form

Suppose $A$ is a $n\times n$ matrix with coefficients in a field , and $\operatorname{rank}(A) + \operatorname{rank}(I_n-A) = n$. How to prove $A^2 = A$ without using Jordan form? What I know : For any $v$ we have $v = Av + (I_n - A )v$. Hence $$V =…
Nicolas Bourbaki
  • 59
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A result for all invertible matrices?

Consider some invertible $N\times N$ matrix $A$, let $B=A^{-1}$ and let $A_{\left[11,nn\right]}$ be the $\left(N-2\right)\times\left(N-2\right)$ matrix resulting from A after removing row and column 1 and row and column $n \neq 1$. I have seen that…
Andres
  • 85
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Quick method /Birds eye view to determine the value

Is there any way to guess the answer without doing elaborating calculations?
user52950
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Basis for $\mathbb{C}^2$

I have doubts about whether \begin{align*} \left\lbrace \begin{pmatrix} \frac{-1-i\sqrt{11}}{2} \\ 1 \end{pmatrix}, \begin{pmatrix} \frac{-1+i\sqrt{11}}{2} \\ 1 \end{pmatrix} \right\rbrace \end{align*} is a basis for $\mathbb{C}^2$. I still do…
Gabriela
  • 850
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Proof of Sylvester's Law of Inertia

I'm getting confused about the proof in my notes of Sylvester's Law of Inertia, and positive definiteness in general. I'll try to state my problems explicitly: The theorem states that if $ \psi : V\times V \to \mathbb{R} $ is a real symmetric…
user938272
  • 923
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Show that $U_1 + U_2 + ... + U_k = \operatorname{span}(U_1 \cup U_2 \cup ... \cup U_k)$

Let us use induction. 1) Base case $k=2$ $U_1 + U_2 = span (U_1 \cup U_2)$, which I understand how to prove is OK (Sum of two subspaces is equal to the span of their union) 2) Induction hypothesis We assume that the following statement holds $$U_1 +…
JD_PM
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