# Tagged Questions

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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### Proof for the distance between a point and a plane

$N$ is a non-zero vector, $c$ a number, and $Q$ a point. $P_0$ is the point of intersection of the line passing through $Q$, in the direction of $N$, and the plane $X\cdot{N} = c$. Show that for all ...
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### Solution space of a linear system

Whats an example of a 2x3 matrix $A$, if possible such that the solution space of the linear system $Ax=0$ is $\mathbb R^3$? I know the zero matrix is one, but does there exist a non-zero matrix?
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### Cofactor expansion to check if matrices is invertible.

I gave question regarding a co-factor expansion question. I understand that an easy way to check if a matrices is invertible is to do co-factor expansion and if $A \ne 0$ then its invertible. I'm ...
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### Equivalent definition of a coset of some subspace of $V$

I am given that $V$ is a vector space over the field $F$, and $X$ is a nonempty subset of $V$ with the following property. $Y=\{x_1-x_2|x_i\in X\}$ is closed under addition and scalar multiplication ...
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### Which matrices are conjugate to an integer valued matrix?

If I have a matrix $A \in M_{n\times n}(\mathbb{C})$, when does there exist a change of basis $B \in Gl_n(\mathbb{C})$ so that $BAB^{-1} \in M_{n\times n}(\mathbb{Z})$? Case $n=1$ is obvious (in ...
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### If A has an eigenvector, a, and A = C+D then a is an eigenvector of C and D

Is the following true? If A has an eigenvector $\vec{a}$ and if A = C+D then $\vec{a}$ is an eigenvector of C and D. Furthermore if A$\vec{a}$ = $\lambda$$\vec{a} and C\vec{a} = \gamma$$\vec{a}$ ...
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### Find all values of 'a' such that given set is a basis of R4?

This is the given set { [1 2 1 3] , [3 4 1 1] , [7 5 a 6] , [1 4 2 a+1] } where we need to find 'a' such that the set is a basis of R4. These are in matrix form with dimension 4 x 1. I proved these ...
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### Linear Algebra - True or False Questions and their Concepts

I have a couple of true-and-false questions whose concepts I need a little help with. 1) If $A$ and $B$ are invertible $n\times n$ matrices, then $(AB)^{-1} = A^{-1}B^{-1}$. 2) If $A$ is a square ...
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### Why does matrix multiplication represent linear transformation compositions?

I know it's probably a silly question, but I'm trying to figure out why was matrix multiplication (the standard one) defined the way it was defined. I know that it was defined like that so we would ...
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### Simplifying Straight Line Equation to plot a Graph. [closed]

I have the equation: 2y-3 = x I need to simplify the 2y to y so I can plot a straight line. How would I do this? If I reorganise the equation I get something like: 2y = x + 3 but if I then ...
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### Basis for series that converge for all complex numbers?

Consider the set of power series that converge on all of $\mathbb C$. Clearly that set forms a vector space. Now I noticed that all functions I know which are in that space (that is, all functions ...
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### Reduced Row Echelon Form

Find all 4x2 matrices in rref form. Attempt at solution. A= \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ ...
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### What does a single eigenvector and eigenvalue for a $2 \times 2$ matrix represent geometrically?

I know that eigenvectors for a matrix $A$ represent lines in the plane that are invariant under the the transformation given by $A$, and the corresponding eigenvalues are the scaling that gets applied ...
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### Inequality - GM, AM, HM and SM means

I've got stuck at this problem : Prove that for any $a > 0$ and any $b > 0$ the following inequality is true:  {3} {\left(\frac{a^3}{b^3} + \frac{b^3}{a^3}\right)} \geq \frac{a}{b} +\...
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### What is the value of the determinant? [duplicate]

\begin{vmatrix} a & b & b & \dots & b \\ b & a & b & \dots & b \\ b & b & a & \dots & b \\ \vdots & \vdots& \vdots & \ddots & \...
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### What is a Spanning Set of vectors? Why do we need Spanning Sets? [closed]

What is a Spanning Set of vectors? What is the use of Spanning Sets in the real world? Please, explain without using format mathematical notations.
The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a $4−by−4$ symmetric positive definite matrix is _____ . I try to explain $:$ Suppose \$...