Question

Show that W, the set of all 3 times 3 upper triangular matrices, forms a subspace of all 3 times 3 matrices. What is the dimension of W? Find a basis for W.

Matrix transformations
ANSWERED
asked 2021-01-25
Show that W, the set of all \(3 \times 3\) upper triangular matrices,
forms a subspace of all \(3 \times 3\) matrices.
What is the dimension of W? Find a basis for W.

Answers (1)

2021-01-26

To show that W, the set of all \(3 \times 3\) upper triangular matrices,
forms asubspace of all \(3 \times 3\) matrices.
Now we know that M, the set of all \(3 \times 3\) matrices, forms a vector space.
Let A, B in W with \(A=\begin{bmatrix}a_{1} & b_{1} & c_{1} \\0 & d_{1} & c_{1}\\ 0 & 0 & f_{1} \end{bmatrix} B=\begin{bmatrix}a_{2} & b_{2} & c_{2} \\0 & d_{2} & c_{2}\\ 0 & 0 & f_{2} \end{bmatrix}\)
and k be a scalar. Now,
\(A+B=\begin{bmatrix}a_{1} & b_{1} & c_{1} \\0 & d_{1} & c_{1}\\ 0 & 0 & f_{1} \end{bmatrix}+\begin{bmatrix}a_{2} & b_{2} & c_{2} \\0 & d_{2} & c_{2}\\ 0 & 0 & f_{2} \end{bmatrix}=\begin{bmatrix}a_{1} + a_{2} & b_{1} +b_{2} & c_{1} + c_{2} \\0 & d_{1}+d_{2} & c_{1}+c_{2}\\ 0 & 0 & f_{1}+f_{2} \end{bmatrix}\)
Once again, \(kA=k\begin{bmatrix}a_{1} & b_{1} & c_{1} \\0 & d_{1} & c_{1}\\ 0 & 0 & f_{1} \end{bmatrix}\)
\(\begin{bmatrix}ka_{1} & kb_{1} & kc_{1} \\0 & kd_{1} & kc_{1}\\ 0 & 0 & kf_{1} \end{bmatrix}\) in W
Therefore, W is a subspace of M. Now let x is a typical element of W, with
\(X=\begin{bmatrix}abc \\0dc\\ 00f \end{bmatrix}\)
can be written as \(A = aE^{11} + bE_{12} + cE_{13} + dE_{22} + cE_{23} + fE_{33}\)
where \(E_{ij}\) is
\(3 \times 3\) time matrix with (i, j) element is 1 and rest are zore.
Now for any linear combination
\(x_{1}E11 + x_{2}E12 + x_{3}E13 + x_{4}E22 + x_{5}E23 + x_{6}E33 = 0_{3 \times 3}\)
imply \(x_{1} = x_{2} = \cdots = x_{6} = 0\)
Therefore, \({E^{11}, E_{12}, E_{13}, E_{22}, E_{23}, E_{33}}\) forms a basis of W,
hence dim \(W = 6.\)

0
 
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2021-06-13
For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A.
\(A=\begin{bmatrix}2&3&5&-9\\-8&-9&-11&21\\4&-3&-17&27\end{bmatrix}\)
Find a nonzero vector in Nul A.
\(A=\begin{bmatrix}-3\\2\\0\\1\end{bmatrix}\)
asked 2021-01-17

Let V be the vector space of real 2 x 2 matrices with inner product
\((A|B) = tr(B^tA)\).
Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for \(U^\perp\) where \(U^{\perp}\left\{A \in V |(A|B)=0 \forall B \in U \right\}\)

asked 2021-05-27
Find k such that the following matrix M is singular.
\(M=\begin{bmatrix}-1 & -1 & -2\\ 0 & -1 & -4 \\ -12+k & -2 & -2 \end{bmatrix}\)
\(k=?\)
asked 2021-06-10
Determine whether the given set S is a subspace of the vector space V.
A. V=\(P_5\), and S is the subset of \(P_5\) consisting of those polynomials satisfying p(1)>p(0).
B. \(V=R_3\), and S is the set of vectors \((x_1,x_2,x_3)\) in V satisfying \(x_1-6x_2+x_3=5\).
C. \(V=R^n\), and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=\(C^2(I)\), and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.
F. V=\(P_n\), and S is the subset of \(P_n\) consisting of those polynomials satisfying p(0)=0.
G. \(V=M_n(R)\), and S is the subset of all symmetric matrices
asked 2021-05-03
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.
\(\begin{bmatrix}2 & 5&5 \\5 & 2&5\\5&5&2 \end{bmatrix}\lambda=-3.12\)
Find P and D
asked 2021-05-19
Use the given inverse of the coefficient matrix to solve the following system
\(5x_1+3x_2=6\)
\(-6x_1-3x_2=-2\)
\(A^{-1}=\begin{bmatrix}-1 & -1 \\2 & \frac{5}{3} \end{bmatrix}\)
asked 2021-01-23
Find basis and dimension \(\displaystyle{\left\lbrace{x}{e}{R}^{{4}}{\mid}{x}{A}={0}\right\rbrace}\) where \(\displaystyle{A}={\left[-{1},{1},{2},{1},{1},{0},{2},{3}\right]}^{{T}}\)
asked 2021-01-15
Let T denote the group of all nonsingular upper triaungular entries, i.e., the matrices of the form, [a,0,b,c] where \(\displaystyle{a},{b},{c}∈{H}\)
\(\displaystyle{H}={\left\lbrace{\left[{1},{0},{x},{1}\right]}∈{T}\right\rbrace}\) is a normal subgroup of T.
asked 2021-01-24
It can be shown that the algebraic multiplicity of an eigenvalue lambda is always greater than or equal to the dimension of the eigenspace corresponding to lambda. Find h in the matrix A below such that the eigenspace for lambda = 5 is two-dimensional: \(\displaystyle{A}={\left[\begin{array}{cccc} {5}&-{2}&{6}&-{1}\\{0}&{3}&{h}&{0}\\{0}&{0}&{5}&{4}\\{0}&{0}&{0}&{1}\end{array}\right]}\)
asked 2021-01-31

(a) Find the bases and dimension for the subspace \(H = \left\{ \begin{bmatrix} 3a + 6b -c\\ 6a - 2b - 2c \\ -9a + 5b + 3c \\ -3a + b + c \end{bmatrix} ; a, b, c \in R \right\}\) (b) Let be bases for a vector space V,and suppose (i) Find the change of coordinate matrix from B toD. (ii) Find \(\displaystyle{\left[{x}\right]}_{{D}}{f}{\quad\text{or}\quad}{x}={3}{b}_{{1}}-{2}{b}_{{2}}+{b}_{{3}}\)

You might be interested in

...