The vector v lies in the subspace of R^3 and is spanned by the set B = {u1, u2}. Making use of the fact that the set B is orthogonal, express v in terms of B where,
v = 1
-2
-13
B = 1 1
2 , 1
3 -1
v is a matrix and B is a set of 2 matrices
To express vector v in terms of the orthogonal set B, we need to find the coefficients of the vectors in B that, when multiplied by the corresponding vectors, will give us v.
Let's call the coefficients of u1 and u2 as c1 and c2, respectively.
To find c1 and c2, we can use the following equation:
v = c1 * u1 + c2 * u2
In this case, we have v = [1, -2, -13], u1 = [1, 2, 3], and u2 = [1, 1, -1].
Now, let's substitute these values into the equation:
[1, -2, -13] = c1 * [1, 2, 3] + c2 * [1, 1, -1]
Expanding the equation, we get:
[1, -2, -13] = [c1, 2c1, 3c1] + [c2, c2, -c2]
Now, equating the corresponding components of both sides of the equation, we have:
1 = c1 + c2
-2 = 2c1 + c2
-13 = 3c1 - c2
We can solve this system of linear equations to find the values of c1 and c2.
Subtracting the first equation from the second equation, we get:
-2 - 1 = (2c1 + c2) - (c1 + c2)
-3 = c1
Substituting the value of c1 back into the first equation, we have:
1 = (-3) + c2
c2 = 1 + 3
c2 = 4
Therefore, c1 = -3 and c2 = 4.
Now, we can substitute these coefficients back into the original equation to express v in terms of B:
v = (-3) * [1, 2, 3] + 4 * [1, 1, -1]
Simplifying the equation, we get:
v = [-3, -6, -9] + [4, 4, -4]
v = [1, -2, -13]
Therefore, the expression of v in terms of the orthogonal set B is:
v = -3u1 + 4u2,
or equivalently:
[1, -2, -13] = -3 * [1, 2, 3] + 4 * [1, 1, -1].