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1. Let A = egin{bmatrix} -3 & -2 & 0 \ 0 & 2 & -6 \ 6 & 3 & 3 end{bmatrix}, and let u = egin{bmatrix} 1 \ 14 \ -9 end{bmatrix} and v = egin{bmatrix} -2 \ 3 \ 1 end{bmatrix}. (a) Determine if u is in Col A. (b) Determine if v is in Nul A. 2. Consider the subset of mathbb{R}^2 consisting of vectors of the form egin{bmatrix} a \ b end{bmatrix}, where a, b in mathbb{R} satisfy a + b = 1. Is this a subspace of mathbb{R}^2? Why or why not? 3. Determine if the following vectors form a basis for mathbb{R}^3. Justify your answer. egin{bmatrix} 1 \ -6 \ -7 end{bmatrix}, egin{bmatrix} 3 \ -4 \ 7 end{bmatrix}, egin{bmatrix} -2 \ 7 \ 5 end{bmatrix}, egin{bmatrix} 0 \ 8 \ 9 end{bmatrix} 4. A matrix A and an echelon form of A are shown below. Find bases for Col A and Nul A, and find the dimensions of these subspaces. A = egin{bmatrix} 1 & -2 & 9 & 5 & 4 \ 1 & -1 & 6 & 5 & -3 \ -2 & 0 & -6 & 1 & -2 \ 4 & 1 & 9 & 1 & -9 end{bmatrix} sim egin{bmatrix} 1 & -2 & 9 & 5 & 4 \ 0 & 1 & -3 & 0 & -7 \ 0 & 0 & 0 & 1 & -2 \ 0 & 0 & 0 & 0 & 0 end{bmatrix} 5. For each of the following constructions, explain why your answer satisfies the required properties. (a) If possible, construct a 4 imes 5 matrix A with rank(A) = 2. (b) If possible, construct a 5 imes 4 matrix A with rank(A) = 3 and dim Nul(A) = 2. 6. Let b_1 = egin{bmatrix} 1 \ -1 \ 3 end{bmatrix}, b_2 = egin{bmatrix} -1 \ 4 \ 1 end{bmatrix}, b_3 = egin{bmatrix} 1 \ -3 \ 1 end{bmatrix}, x = egin{bmatrix} 1 \ 2 \ 3 end{bmatrix}. The set mathcal{B} = { b_1, b_2, b_3 } forms a basis for mathbb{R}^3. Find [x]_mathcal{B}, the mathcal{B}-coordinate vector of x.

          1. Let A = egin{bmatrix} -3 & -2 & 0 \ 0 & 2 & -6 \ 6 & 3 & 3 end{bmatrix}, and let u = egin{bmatrix} 1 \ 14 \ -9 end{bmatrix} and v = egin{bmatrix} -2 \ 3 \ 1 end{bmatrix}.
(a) Determine if u is in Col A.
(b) Determine if v is in Nul A.
2. Consider the subset of mathbb{R}^2 consisting of vectors of the form egin{bmatrix} a \ b end{bmatrix}, where a, b in mathbb{R} satisfy a + b = 1. Is this a subspace of mathbb{R}^2? Why or why not?
3. Determine if the following vectors form a basis for mathbb{R}^3. Justify your answer.
egin{bmatrix} 1 \ -6 \ -7 end{bmatrix}, egin{bmatrix} 3 \ -4 \ 7 end{bmatrix}, egin{bmatrix} -2 \ 7 \ 5 end{bmatrix}, egin{bmatrix} 0 \ 8 \ 9 end{bmatrix}
4. A matrix A and an echelon form of A are shown below. Find bases for Col A and Nul A, and find the dimensions of these subspaces.
A = egin{bmatrix} 1 & -2 & 9 & 5 & 4 \ 1 & -1 & 6 & 5 & -3 \ -2 & 0 & -6 & 1 & -2 \ 4 & 1 & 9 & 1 & -9 end{bmatrix} sim egin{bmatrix} 1 & -2 & 9 & 5 & 4 \ 0 & 1 & -3 & 0 & -7 \ 0 & 0 & 0 & 1 & -2 \ 0 & 0 & 0 & 0 & 0 end{bmatrix}
5. For each of the following constructions, explain why your answer satisfies the required properties.
(a) If possible, construct a 4 	imes 5 matrix A with rank(A) = 2.
(b) If possible, construct a 5 	imes 4 matrix A with rank(A) = 3 and dim Nul(A) = 2.
6. Let
b_1 = egin{bmatrix} 1 \ -1 \ 3 end{bmatrix}, b_2 = egin{bmatrix} -1 \ 4 \ 1 end{bmatrix}, b_3 = egin{bmatrix} 1 \ -3 \ 1 end{bmatrix}, x = egin{bmatrix} 1 \ 2 \ 3 end{bmatrix}.
The set mathcal{B} = { b_1, b_2, b_3 } forms a basis for mathbb{R}^3. Find [x]_mathcal{B}, the mathcal{B}-coordinate vector of x.

1. Let A = eginbmatrix -3 -2 0 0 2 -6 6 3 3 endbmatrix, and let u = eginbmatrix 1 14 -9 endbmatrix and v = eginbmatrix -2 3 1 endbmatrix.
(a) Determine if u is in Col A.
(b) Determine if v is in Nul A.
2. Consider the subset of mathbbR^2 consisting of vectors of the form eginbmatrix a b endbmatrix, where a, b in mathbbR satisfy a + b = 1. Is this a subspace of mathbbR^2? Why or why not?
3. Determine if the following vectors form a basis for mathbbR^3. Justify your answer.
eginbmatrix 1 -6 -7 endbmatrix, eginbmatrix 3 -4 7 endbmatrix, eginbmatrix -2 7 5 endbmatrix, eginbmatrix 0 8 9 endbmatrix
4. A matrix A and an echelon form of A are shown below. Find bases for Col A and Nul A, and find the dimensions of these subspaces.
A = eginbmatrix 1 -2 9 5 4 1 -1 6 5 -3 -2 0 -6 1 -2 4 1 9 1 -9 endbmatrix sim eginbmatrix 1 -2 9 5 4 0 1 -3 0 -7 0 0 0 1 -2 0 0 0 0 0 endbmatrix
5. For each of the following constructions, explain why your answer satisfies the required properties.
(a) If possible, construct a 4 imes 5 matrix A with rank(A) = 2.
(b) If possible, construct a 5 imes 4 matrix A with rank(A) = 3 and dim Nul(A) = 2.
6. Let
b1 = eginbmatrix 1 -1 3 endbmatrix, b2 = eginbmatrix -1 4 1 endbmatrix, b3 = eginbmatrix 1 -3 1 endbmatrix, x = eginbmatrix 1 2 3 endbmatrix.
The set mathcalB = b1, b2, b3 forms a basis for mathbbR^3. Find [x]mathcalB, the mathcalB-coordinate vector of x.

Added by Jose Carlos R.

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