Our exercise is to find all solutions to the equation $Ax = 0$, among others for the following matrix $$A =\begin{pmatrix} 6 & 3 & -9 \\ 2 & 1 & -3 \\ -4 & -2 & 6 \end{pmatrix}.$$ This amounts to finding the kernel, and obviously, the rows of the matrix are multiples of […]

I know that for every symmetric form $f: U \times U \rightarrow \mathbb{K}$, char$\mathbb{K} \neq 2$ there exists a basis for which $f$’s matrix is diagonal. Could you tell me what happens if we omit assumption about $f$ being symmetric? Could you give me an example of non symmetric bilinear form $f$ which cannot be […]

Well, I was reading about invariant subspaces and related things and this question came to my mind: If I choose a vector space and fix a linear transformation on itself, then how many invariant subspaces will there be? Is there any formula or materials to read?

As the title, let $(V,\langle,\rangle)$ be a complex inner product space and assume $S_1=(u_1,\ldots,u_n)$, $S_2=(v_1,\ldots,v_n)$ are orthonormal bases of $V$. Prove that the change of basis matrix $M_ IV(S_2,S_1)$ is a unitary matrix. (There is a hint that let $S$ be the operator s.t. $S(u_i)=v_i$ and prove this is a unitary operator.)

Suppose $A$ is an invertible square matrix and $u,v$ are column vectors. Suppose furthermore that $1 + v^T A^{-1}u \neq 0$. Then the Sherman–Morrison formula states that: $$ (A+uv^T)^{-1} = A^{-1} – {A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u} $$ Here, $uv^T$ is the outer product of two vectors $u$ and $v$. A proof I […]

Ax = b. I need a way to analyze a square matrix A to see if its solution vector x will always be positive when b is positive. This question arises from solving the radiosity equation: I’m interested to know when A is incorrect, which would be when x has negative values even though b […]

A matrix $a\in GL_{n}(F)$ is said to be monomial if each row and column has exactly one non-zero entry. Let $N$ denote the set of all monomial matrices. I want to prove that following are equivalent $A\in N$ there exist a non singular $D$ (diagonal matrix ) and a permutation matrix $P$ such that $A=DP$ […]

Problem : If both $A-\frac{1}{2}I$ and $A + \frac{1}{2}I$ are orthogonal matrices, then which one of the following is correct : (i) A is orthogonal (2) A is skew symmetric matrix of even order (3) $A^2 = \frac{3}{4}I$ Solution : $(A’-\frac{1}{2}I)(A-\frac{1}{2}I) =I$ and $(A’+\frac{1}{2}I)(A+\frac{1}{2}I) =I$ $\Rightarrow A +A’ =0$ $\Rightarrow A’ =-A $ $\Rightarrow A^2 […]

Let $K$ be a field. Consider the vector space $K^\Bbb{N}$ of $K$-sequences. Is there an uncountable linearly independent set of vectors in this vector space? If Yes, can you name it explicitely? Does this work for modules as well?

Let say I have this figure, I know slope $m_1$, slope $m_1$, $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$. I need to calculate slope $m_3$. Note the line with $m_3$ slope will always equally bisect line with $m_1$ slope and line with $m_2$.

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