### Vector and Matrix

Exercise Draw the vectors (1, 3), (0, 1), (0, 0), (-1, -3), (2, -2).

Exercise Draw the collections of vectors.

X = {(0, y): y is any number}

Y = {(x, 0): x is any number}

S = {(x, y): x2 + y2 = 1}

D = {(x, x): x is any number}

E = {(x, y): x + 3y = 0}

Exercise Let u = (1,-5), v = (2, 4 ,1), w = (3, 0, 1, -1, 4), x = (2, -4, 3), y = (-7, 2, 1), z = (1, 2, 0, -3, 4). Compute.

u + v, v + y, y + v, w + z, z + w, x + y + v,

2u, -3v, -w, 0x,

2u + 5v, -2y + 3v, w - 3z, 4x + 2y + v.

Exercise Compute.

 [ 1 ] - [ 5 ], [ 2 ] - [ 7 ], 3[ 1 ] - 5[ 5 ], 3[ 2 ] - 5[ 7 ] 3 6 4 8 3 6 4 8
 [ 1 2 ] - [ 5 7 ], 3[ 1 2 ] - 5[ 5 7 ] 3 4 6 8 3 4 6 8
 [ 0 -2 3 ] + [ 2 0 -1 ], -2[ 0 -2 3 ] + 3[ 2 0 -1 ] 1 0 -3 0 4 -3 1 0 -3 0 4 -3 2 -2 2 3 -3 0 2 -2 2 3 -3 0
 a[ 1 0 0 ] + b[ 0 0 0 ] + c[ 0 0 0 ] 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
 a11[ 1 0 ] + a12[ 0 1 ] + a21[ 0 0 ] + a22[ 0 0 ] 0 0 0 0 1 0 0 1
 [ 1 -3 5 10 ] + [ 4 -9 8 ], 12[ ] - 13[ 0 0 0 0 0 ] 0 5 1 -7 4 0 3 0 0 0 0 0 0 0 0 0 0 1 0 9 -3 1 4 0 0 0 0 0 0 0 0 0 0 0 2 0 1 0 0 0 4 0 0 0 0 0 0 0 0 0 0 -6 3 0 0 0 5 3 0 0 0 0 0

Exercise Take the standard basis vectors e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) of R3. Compute the linear combination x1e1 + x2e2 + x3e3. Then generalize your result to other dimensions.

Exercise For u = (1, 2), v = (3, 4), w = (5, 6), compute the linear combination x1u + x2v + x3w. Do the same for a1 = (1, 2, 3), a2 = (4, 5, 6), a3 = (7, 8, 9). Express your computation in vertical vectors and draw a general observation.

Exercise Express the vectors b = (4, 6), c = (1, 1), e1 = (1, 0), e2 = (0, 1) as linear combinations of the vectors u = (1, 2), v = (3, 4), w = (5, 6). Can you express as linear combinations of the vectors u, v only?

Exercise Express the vectors b = (1, 1, 1), c = (-1, 0, 1), e1 = (1, 0, 0) as linear combinations of the vectors a1 = (1, 2, 3), a2 = (4, 5, 6), a3 = (7, 8, 9). Is the expression (i.e., the coefficients) unique?