Question

Problem 12. (7 points) Determine if the statements are true or false. 1. Any four vectors in $mathbb{R}^3$ are linearly dependent. ? 2. Any four vectors in $mathbb{R}^3$ span $mathbb{R}^3$. ? 3. The rank of a matrix is equal to the number of pivots in its RREF. ? 4. {$v_1, v_2, dots, v_n}$ is a basis for span($v_1, v_2, dots, v_n$). ? 5. If $v$ is an eigenvector of a matrix $A$, then $v$ is an eigenvector of $A + cI$ for all scalars $c$. (Here $I$ denotes the identity matrix of the same dimension as $A$.) ? 6. An $n imes n$ matrix $A$ is diagonalizable if and only if it has $n$ distinct eigenvalues. ? 7. Let $W$ be a subspace of $mathbb{R}^n$. If $p$ is the projection of $b$ onto $W$, then $b - p in W^perp$.?

          Problem 12.
(7 points)
Determine if the statements are true or false.
1. Any four vectors in $mathbb{R}^3$ are linearly dependent. ?
2. Any four vectors in $mathbb{R}^3$ span $mathbb{R}^3$. ?
3. The rank of a matrix is equal to the number of pivots in its RREF. ?
4. {$v_1, v_2, dots, v_n}$ is a basis for span($v_1, v_2, dots, v_n$). ?
5. If $v$ is an eigenvector of a matrix $A$, then $v$ is an eigenvector of $A + cI$ for all scalars $c$. (Here $I$ denotes the identity matrix of the same dimension as $A$.) ?
6. An $n 	imes n$ matrix $A$ is diagonalizable if and only if it has $n$ distinct eigenvalues. ?
7. Let $W$ be a subspace of $mathbb{R}^n$. If $p$ is the projection of $b$ onto $W$, then $b - p in W^perp$.?

Problem 12.
(7 points)
Determine if the statements are true or false.
1. Any four vectors in mathbbR^3 are linearly dependent. ?
2. Any four vectors in mathbbR^3 span mathbbR^3. ?
3. The rank of a matrix is equal to the number of pivots in its RREF. ?
4. v1, v2, dots, vn is a basis for span(v1, v2, dots, vn). ?
5. If v is an eigenvector of a matrix A, then v is an eigenvector of A + cI for all scalars c. (Here I denotes the identity matrix of the same dimension as A.) ?
6. An n imes n matrix A is diagonalizable if and only if it has n distinct eigenvalues. ?
7. Let W be a subspace of mathbbR^n. If p is the projection of b onto W, then b - p in W^perp.?

Added by Luc-A T.

Question

Please give Ace some feedback