Simplifying Linear Algebra: Techniques, Solutions, and Examples

Determine the eigenvalues, eigenveltI and eigenspace of the follewing matix
${\displaystyle \left[\begin{array}{cc}1& 3\\ 5& 3\end{array}\right]}$

Let V and W be vector spaces, let ${\displaystyle T:V\to W}$ be linear, and let ${\displaystyle \{w_1,w_2,\dots ,w_k\}}$ be a linearly independent set of k vectors from R(T). Prove that if ${\displaystyle S=\{v_1,v_2,...,v_k\}}$ is chosen so that ${\displaystyle T\left(v_i\right)=W_i}$ for ${\displaystyle i=1,2,\dots ,k,}$ then S is linearly independent.

Find the distance between two planes:
${\displaystyle C_1:x+y+2z=4}$ and ${\displaystyle C_2:3x+3y+6z=18}$
And find the other plane ${\displaystyle C_3\ne C_1}$ that has the distance d po the plain ${\displaystyle C_2}$

Assume that T is a linear transformation. Find the standard matrix of T.
T: ${\displaystyle {\mathbb{R}}^2\to {\mathbb{R}}^4,T\left(e_1\right)=(3,1,3,1),\text{and}T\left(e_2\right)=(-5,6,0,0)}$, where ${\displaystyle e_1=(1,0)}$ and ${\displaystyle e_2=(0,1)}$
${\displaystyle A=}$

What is the geometric meaning of singular matrix
Could anyone help explain what is the geometric meaning of singular matrix ? Whats

How can I find the dimension of the eigenspace?
The matrix $$A=\left[\begin{array}{cc}9& -1\\ 1& 7\end{array}\right]$$ has one eigenvalue of multiplicity 2. Find this eigenvalue and the dimension of the eigenspace.
So we found the eigenvalue by doing ${\displaystyle A-\lambda I}$ to get:
${\displaystyle \lambda =8}$
But how exactly do we find the dimension of the eigenspace?

Expressing the determinant of a sum of two matrices?
Can ${\displaystyle det(A+B)}$
be expressed in terms of
${\displaystyle det\left(A\right),det\left(B\right),n}$
where A,B are ${\displaystyle n\times n}$ matrices?
I made the edit to allow n to be factored in.

Can someone show me step-by-step how to diagonalize this matrix? Im

Find k such that the following matrix M is singular.
${\displaystyle \left[\begin{array}{ccc}-3& -3& 3\\ 6& 7& -6\\ 11+k& 16& -15\end{array}\right]}$

Physical meaning of the null space of a matrix
What is an intuitive meaning of the null space of a matrix? Why is it useful?
I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it.
E.g.: I think of the rank r of a matrix as the minimum number of dimensions that a linear combination of its columns would have; it tells me that, if I combined the vectors in its columns in some order, I'd get a set of coordinates for an r-dimensional space, where r is minimum (please correct me if I'm wrong). So that means I can relate rank (and also dimension) to actual coordinate systems, and so it makes sense to me. But I can't think of any physical meaning for a null space... could someone explain what its meaning would be, for example, in a coordinate system?

f(x)=${\displaystyle \int \sqrt{1-x^2\phantom{!}}dx}$

Is division of matrices possible?
Is it possible to divide a matrix by another? If yes, What will be the result of $\frac{A}{B}$ if
$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

$B=\left(\begin{array}{cc}w& x\\ y& z\end{array}\right)?$

How to find absolute value of a vector in this question?
a = {1, 3, 5}
Then in formulas see this |a|

Is a matrix multiplied with its transpose something special?
In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.
Is ${\displaystyle {\mathrm{\forall }}^T}$ something special for any matrix A?

How do you find the scalar and vector projections of b onto a given ${\displaystyle a=i+j+k}$, ${\displaystyle b=i-j+k}$?

Find an equation for the plane that
(a) is perpendicular to ${\displaystyle \text{f{v} = (1, 1, 1}f\left\{v\right\}=(1,1,1)}$ and passes through (1,0,0).
(b) is perpendicular to ${\displaystyle \text{f{v} = (1, 2, 3}f\left\{v\right\}=(1,2,3)}$ and passes through (1,1,1).
(c) is perpendicular to the line ${\displaystyle \text{f{l}(t) = (5, 0, 2)t + (3, −1, 1}f\left\{l\right\}\left(t\right)=(5,0,2)t+(3,-1,1)}$ and passes through (5, −1, 0)(5,−1,0).
(d) is perpendicular to the line ${\displaystyle \text{f{l}(t) = (−1, −2, 3)t + (0, 7, 1}f\left\{l\right\}\left(t\right)=(-1,-2,3)t+(0,7,1)}$ and passes through (2, 4, −1)(2,4,−1).

Let ${\displaystyle a_1=\left[\begin{array}{c}1\\ 3\\ -1\end{array}\right],a_2=\left[\begin{array}{c}-6\\ -14\\ 3\end{array}\right],b=\left[\begin{array}{c}5\\ 3\\ h\end{array}\right]}$. for what value(s) of h is b in the plane spanned by ${\displaystyle a_1}$ and ${\displaystyle a_2}$

Find the characteristic equation and the eigenvalues (and corresponding eigenvectors) of the matrix.
$$\left[\begin{array}{ccc}2& -2& 3\\ 0& 3& -2\\ 0& -1& 2\end{array}\right]$$
a) The characteristic equation
${\displaystyle {\lambda }^3-7{\lambda }^2+14\lambda -8=0}$
b) The eigenvalues
${\displaystyle ({\lambda }_1,{\lambda }_2,{\lambda }_3)=(1,2,4)}$
The corresponding eigenvectors
${\displaystyle x_1=-7,1,1?}$
${\displaystyle x_2=1,0,0}$
${\displaystyle x_3=13,-4,2?}$

Find, correct to the nearest degree, the three angles of the triangle with the given vertices.
${\displaystyle A(1,0,-1),B(3,-4,0),C(1,3,5)}$