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): x^{2} + y^{2} = 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 |
a_{11}[ | 1 | 0 | ] + a_{12}[ | 0 | 1 | ] + a_{21}[ | 0 | 0 | ] + a_{22}[ | 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 e_{1} = (1, 0, 0), e_{2} = (0, 1, 0), e_{3} = (0, 0, 1) of R^{3}. Compute the linear combination x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3}. Then generalize your result to other dimensions.
Exercise For u = (1, 2), v = (3, 4), w = (5, 6), compute the linear combination x_{1}u + x_{2}v + x_{3}w. Do the same for a_{1} = (1, 2, 3), a_{2} = (4, 5, 6), a_{3} = (7, 8, 9). Express your computation in vertical vectors and draw a general observation.
Exercise Express the vectors b = (4, 6), c = (1, 1), e_{1} = (1, 0), e_{2} = (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), e_{1} = (1, 0, 0) as linear combinations of the vectors a_{1} = (1, 2, 3), a_{2} = (4, 5, 6), a_{3} = (7, 8, 9). Is the expression (i.e., the coefficients) unique?
Exercise Express the matrix
X = [ | 3 | 4 | ] |
2 | 3 |
as linear combinations of the matrices.
A = [ | 1 | 2 | ], B = [ | 1 | 0 | ], C = [ | 0 | 1 | ] |
0 | 1 | 2 | 1 | -1 | 0 |
Is the expression (i.e., the coefficients) unique? Is it true that any 2 by 2 matrix is a linear combination of A, B and C?