WOLFRAM|DEMONSTRATIONS PROJECT

Sets of Linear Combinations and Their Images under Linear Transformations

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matrix A
A
11
0
A
12
0
A
13
0
A
21
0
A
22
0
A
23
0
A
31
0
A
32
0
A
33
0
r
1
scalars for vector a
0
r
2
scalars for vector b
0
r
3
scalars for vector c
0
vector a
a
1
5
a
2
0
a
3
0
vector b
b
1
0
b
2
7
b
3
0
vector c
c
1
0
c
2
0
c
3
11
point p
p
1
0
p
2
0
p
3
0
predetermined vector u
predetermined vector v
This Demonstration visualizes points, vectors, and the effect of linear transformations from
3
R
to
3
R
. The black and red vectors are linear combinations of
a
and
b
given by
3a+2b
and
2a-b+c
, respectively. The green dots are integral scalar multiples of
a,b
, and
c
, and the black dots show the images of the green dots under the linear transformation defined by the matrix
A
.