Practice Final #4
Cullen_Im
a. angle = 108.317 degrees. b. inner product space is dot product of vectors [a + 3b, a + b] and [c + 3d, c + d]
Cullen_Im: May 29, 2015, 8:04 p.m.
Because [a,b] with respect to your friend's basis is [a + 3b, a + b] in the standard basis and [c,d] w.r.t. your friend's basis is [c + 3d, c + d] in the standard basis, the inner product is the dot product of the two vectors
tedallen: May 30, 2015, 9:12 p.m.
I think Prof. Weisbart includes an additional absolute value around $(v \cdot w)$ in the formula for part a. In that case the answer would be $\theta = cos^{-1}\Big( \frac{2\sqrt{2}}{9} \Big)
tedallen: May 30, 2015, 9:12 p.m.
I think Prof. Weisbart includes an additional absolute value around $(v \cdot w)$. In that case the answer would be $\theta = cos^{-1}\Big( \frac{2\sqrt{2}}{9} \Big)$ (sorry about posting twice, the formatting failed)
cwhiteside7: May 31, 2015, 5:42 p.m.
For the inner product equation, should you be using different variables instead of a and b (ex. [a+3b,a+b]) since these are simply constants that will change the basis?
yulduzkhonbruin: June 1, 2015, 1:37 p.m.
tedallen, I checked the edited practice finals and I do not see the absolute values around the vectors. Which one was it from? But, for this question, I got the same answer as Cullen_Im.
aleksanderjanczewski: June 1, 2015, 11:25 p.m.
I got different vectors than you, I guess you shoud try to derive identity matrix for change of basis and then multiply vector <a,b> and <c,d> by this matrix. This will give you these vectors with respect to your friend's basis.
TLindberg: June 2, 2015, 4:39 p.m.
A more concise way to state: $\langle \begin{pmatrix} a \\ b \end{pmatrix} , \begin{pmatrix} c \\ d \end{pmatrix}\rangle$ is 2ac + 4bc + 4ad + 10bd. $\\$ You can do this by computing the dot product of the the two vectors $\begin{pmatrix} a+ 3b \\ a + b \end{pmatrix} \bullet \begin{pmatrix} c + 3d \\ c + d \end{pmatrix}\\$ I'm not sure if this is better, but this might be a safer answer on the test, because it shows that you more thoroughly understand the concept.
Cullen_Im: June 3, 2015, 12:27 a.m.
@tedallen: I don't believe there is a an absolute value around (v dot product w) because you're trying to find the angle between vectors and not between lines. The absolute value makes the angle be less than 90 degrees, but the angle between vectors can be greater than 90 degrees so the absolute value is not needed.