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Is the matrix 1 0, 0 1
in the span of the following three matrices: 3 1, 1 -1,
0 1, 1 2,
1 1, 1 0
In order to answer this question,
we need to determine if we can write this matrix here as a linear combination
of these three matrices.
In other words, we want to solve
this equation for the scalars
a, b, and c. And if there's no solution,
then we can conclude that the answer is "no".
Otherwise, we can say that this matrix
is in the span of the three matrices
So let's expand this.
The right-hand side can be simply written
as the following matrix.
So I have 3a + 0b + c in the top-left entry,
and for the top-right entry I'm going to have a + b + c
and in the bottom left I have
a + b + c
and in the bottom right I have -a + 2b + 0c.
And now we want to solve for a, b, and c.
So we just compare the corresponding entries.
So I have 1 = 3a + c
and I have 0 = a + b + c
and 0
equals a + b + c, that's an equation that we already have.
And finally 1 = -a + 2b.
So let me label these equations (1), (2), and (3).
Now if we add -2 times the second equation
to the third equation, we get the following:
-3 a - 2 c = 1.
And let's call that (4).
Now (1) + (4) gives us -c = 2.
So c is -2.
And if we substitute this back into (1)
I get a = 1
And if a equals 1
the second equation tells us that b must be 1 as well.
So that means
the answer to this question is "yes" because we can write
1 0, 0 1 as a linear combination of
these three matrices here.