Question

The vector x is in a subspace H with a basis Bequals​{Bold b 1​,Bold b 2​}. Find the​ B-coordinate vector of x. Bold b 1equals[Start 3 By 1 Matrix 1st Row 1st Column 1 2nd Row 1st Column 2 3rd Row 1st Column negative 3 EndMatrix ]​, Bold b 2equals[Start 3 By 1 Matrix 1st Row 1st Column negative 4 2nd Row 1st Column negative 7 3rd Row 1st Column 11 EndMatrix ]​, xequals[Start 3 By 1 Matrix 1st Row 1st Column negative 10 2nd Row 1st Column negative 17 3rd Row 1st Column 27 EndMatrix ]

Answer 1

Answer and Step-by-step explanation: To find the B-coordinate vector of x:

`b_(1) = \left[\begin{array}{ccc}1\\2\\-3\end{array}\right]` ,
`b_(2) = \left[\begin{array}{ccc}-4\\-7\\11\end{array}\right]`, x =
`\left[\begin{array}{ccc}-10\\-17\\27\end{array}\right]`

The augmented matrix will be:

`\left[\begin{array}{ccc}1&-4&-10\\2&-7&-17\\-3&11&27\end{array}\right]`

Transforming into reduced row-echelon form:

=
`\left[\begin{array}{ccc}1&-4&-10\\0&1&3\\0&-1&-3\end{array}\right]` =
`\left[\begin{array}{ccc}1&-4&-10\\0&1&3\\0&0&0\end{array}\right]`

=
`\left[\begin{array}{ccc}1&0&2\\0&1&3\\0&0&0\end{array}\right]`

The values for the vector will be:

x = 2

y = 3

The B-coordinate vector is of the form:

V =
`\left[\begin{array}{ccc}x\\y\end{array}\right]`

V =
`\left[\begin{array}{ccc}2\\3\end{array}\right]`

The B-coordinate vector of x is V =
`\left[\begin{array}{ccc}2\\3\end{array}\right]`