Subspace of the vector space of nxn matrices, \(\displaystyle{M}_{{{n}{x}{n}}}\) are the subsets of the diagonal matrices and upper triangular matrices.

Jayden-James Duffy

Answered 2021-10-14
Author has **26785** answers

asked 2021-12-11

Which of the subsets of \(\displaystyle{R}^{{{3}×{3}}}\) are subspaces of \(\displaystyle{R}^{{{3}×{3}}}\)?

a) The invertible \(\displaystyle{3}\times{3}\) matrices

b) The \(\displaystyle{3}\times{3}\) matrices whose entries are all integers

c) The \(\displaystyle{3}\times{3}\) matrices with all zeros in the third row

d) The non-invertible \(\displaystyle{3}\times{3}\) matrices

e) The diagonal \(\displaystyle{3}\times{3}\) matrices

f) The symmetric \(\displaystyle{3}\times{3}\) matrices

a) The invertible \(\displaystyle{3}\times{3}\) matrices

b) The \(\displaystyle{3}\times{3}\) matrices whose entries are all integers

c) The \(\displaystyle{3}\times{3}\) matrices with all zeros in the third row

d) The non-invertible \(\displaystyle{3}\times{3}\) matrices

e) The diagonal \(\displaystyle{3}\times{3}\) matrices

f) The symmetric \(\displaystyle{3}\times{3}\) matrices

asked 2022-01-04

(a) Suppose U and W are subspaces of a vector space V. Prove that \(\displaystyle{U}\cap{W}={\left\lbrace{v}:{v}\in{U}\text{and}\ {v}\in{W}\right\rbrace}\) is a subspace of V.

(b) Give an example of two subspaces U and W and a vector space V such that \(\displaystyle{U}\cup{W}={\left\lbrace{v}:{v}\in{U}\text{or}\ {v}\in{W}\right\rbrace}\) is not a subspace of V.

(b) Give an example of two subspaces U and W and a vector space V such that \(\displaystyle{U}\cup{W}={\left\lbrace{v}:{v}\in{U}\text{or}\ {v}\in{W}\right\rbrace}\) is not a subspace of V.

asked 2021-12-08

Tell, please, what are the \(\displaystyle\prec\) and \(\displaystyle\succ\) operators for when you use it with matrices?

asked 2021-11-28

Find a basis for the space of \(\displaystyle{2}\times{2}\) lower triangular matrices

Basis =

Basis =

asked 2021-05-14

Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The function are real-valued. All rational functions f/g, with the degree off ≤≤ the degree ofg (including f = 0).

asked 2021-11-08

Find the value(s) of h for which the vectors are linearly dependent. Justify each answer.

\[\begin{bmatrix}1 \\5 \\-3\end{bmatrix},\begin{bmatrix}-2 \\-9 \\6\end{bmatrix},\begin{bmatrix}3 \\h \\-9\end{bmatrix}\]

\[\begin{bmatrix}1 \\5 \\-3\end{bmatrix},\begin{bmatrix}-2 \\-9 \\6\end{bmatrix},\begin{bmatrix}3 \\h \\-9\end{bmatrix}\]

asked 2021-11-09

Describe all solutions of Ax=0 in parametric
vector form, where A is row equivalent to the given matrix.

\[\begin{bmatrix}1 & 3 & 0 & -4\\2 & 6 & 0 & -8\end{bmatrix}\]

\[\begin{bmatrix}1 & 3 & 0 & -4\\2 & 6 & 0 & -8\end{bmatrix}\]