Answer to Question #235241 in Algebra for Ash

Question #235241

1). Determine whether set S given below is a basis for ℝ 3 . If not, explain why.

S = {(1,0,0),(0,1,0),(0,0,0)}


2). Find the rank of matrix D given below.

D = [0 3 9 0]

[-2 -1 1 -1]

[0 -1 -3 1]


1
Expert's answer
2021-09-16T00:43:30-0400

a) A subset SS of a vector space VV is called a basis if

1. SS is linearly independent, and

2. SS  is a spanning set

Any three linearly independent vectors form a basis of R3.\R^3.

Let us check that whether SS is a linearly independent set.

Consider the linear combination


x1(100)+x2(010)+x3(000)=(000)x_1\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}+x_2\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}+x_3\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}=\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}

This is equivalent to the matrix equation


(100010000)(x1x2x3)=(000)\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}=\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}A=(100001000000)A=\begin{pmatrix} 1 & 0 & 0 & & 0 \\ 0 & 1 & 0 & & 0 \\ 0 & 0 & 0 & & 0 \\ \end{pmatrix}

Thus, the general solution is x1=0,x2=0,x3=t,tRx_1=0, x_2=0, x_3=t, t\in \R is a free variable.

Hence, in particular, there is a nonzero solution.

So SS is linearly dependent, and hence SS cannot be a basis for R3.\R^3.


b)


D=[039021110131]D= \begin{bmatrix} 0 & 3 & 9 & 0 \\ -2 & -1 & 1 & -1 \\ 0 & -1 & -3 & 1 \\ \end{bmatrix}

Swap R1R_1 and R2R_2


[211103900131]\begin{bmatrix} -2 & -1 & 1 & -1 \\ 0 & 3 & 9 & 0 \\ 0 & -1 & -3 & 1 \\ \end{bmatrix}

R3=R3+R2/3R_3=R_3+R_2/3


[211103900001]\begin{bmatrix} -2 & -1 & 1 & -1 \\ 0 & 3 & 9 & 0 \\ 0 & 0 & 0& 1 \\ \end{bmatrix}

The rank of a matrix is the number of nonzero rows in the reduced matrix, so the rank is 3.


rank(D)=3rank(D)=3


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