Questions asked
10) Is the given set of vectors a vector space? Give reasons. If your answer is yes, determine the dimension and find abasis. \( \left(a_{1}, a_{2}, a_{3}, a_{4} \ldots .\right. \). denote components.) All vectors in with (a) All vectors in \( R^{3} \) with \( a_{1}-a_{2}+2 a_{3}=0 \). (b) All vectors in \( R^{3} \) with \( 3 a_{2}+a_{3}=k \). (c) All vectors in \( R^{n} \) with the first \( n-2 \) components zero.
Mukesh Devi
Numerade educator
9) Are there values of r and s for which rank is 2? ?1 3 -1 ? ?0 r^2 - 2 2 ? ?0 s - 11 r + 2? ?0 0 2 ?
8) Find the rank. Find a basis for the row space. Find a basis for the column space. \[ \left[\begin{array}{ccc} 2 & -1 & 4 \\ 0 & 11 & -3 \\ 1 & 0 & 21 \end{array}\right] \]
Willis James
For what value of k, the vector (1, k, -2) is a linear combination of e1 = (1,2,3) and e2 = (2,3,1)?
Hoan Nguyen
The set ? = {(1,2,3), (1,0,1) , (0,1,0), (0,0,1)} is a linearly dependent subset of the vector space R3(R).
3) Show that the set of all elements of the form \( (x+2 y, x+3 z, 2 x+y+z) \), where \( x, y, z \in F \), is a subspace of \( V^{3}(F) \)
1) Let \( \mathrm{V} \) be the set of all ordered pairs \( (x, y) \) of real numbers and \( F \) be the field of real numbers. Define \[ (x, y)+\left(x_{1}, y_{2}\right)=\left(3 y+3 y_{1},-x-x_{1}\right) \text { and } c(x, y)=(3 c y,-c x) \] Verify that \( V \), with these operations, is not a vector space over the field of real numbers.
5. Find the basis for the eigenspace corresponding to the given eigenvalues, (a) \[ A=\left[\begin{array}{ll} 3 & 0 \\ 2 & 1 \end{array}\right], \lambda=1,3 \] (b) \[ A=\left[\begin{array}{rrr} -4 & 1 & 1 \\ 2 & -3 & 2 \\ 3 & 3 & -2 \end{array}\right], \lambda=-5 \]
4. Let \[ A=\left[\begin{array}{rrr} 1 & -2 & 8 \\ 0 & -1 & 0 \\ 0 & 0 & -3 \end{array}\right] \text { and } P=\left[\begin{array}{rrr} 1 & -4 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right] \] Confirm that \( P \) diagonalizes \( A \), and then compute each of the following powers of \( A \). (a) \( A^{1000} \) (b) \( A^{-1000} \) (c) \( A^{2301} \) (d) \( A^{-2301} \)