# Find the standard inner product on P^2 of the given polynomials p=−5+4x+x^2,q=3+4x–2x^2.

Question
Matrix transformations
Find the standard inner product on $$\displaystyle{P}^{{2}}$$ of the given polynomials $$\displaystyle{p}=−{5}+{4}{x}+{x}^{{2}},{q}={3}+{4}{x}–{2}{x}^{{2}}$$.

2020-11-21
On $$\displaystyle{P}^{{2}}$$ the standart inner product is given by, for $$\displaystyle{p}{\left({x}\right)}={a}{0}+{a}{1}{x}+{a}{2}{x}^{{2}},{q}{\left({x}\right)}={b}{0}+{b}{1}{x}+{b}{2}{x}^{{2}}∈{P}^{{2}}$$.
PSK(p,q)=a0b0+a1b1x+a2b2x^2 =-5*3+4*4x+(-2)x^2 =-15+16x-2x^2ZSK

### Relevant Questions

Use the weighted Euclidean inner product on R2 ‹u, v› = 99u1v1 + 5u2v2 where u = (u1, u2) and v = (v1, v2), to find ||w||, where w = (− 1, 3).
Give the correct answer and solve the given equation
Let $$\displaystyle{p}{\left({x}\right)}={2}+{x}{\quad\text{and}\quad}{q}{\left({x}\right)}={x}$$. Using the inner product $$\langle\ p,\ q\rangle=\int_{-1}^{1}pqdx$$ find all polynomials $$\displaystyle{r}{\left({x}\right)}={a}+{b}{x}\in{P}{1}{\left({R}\right)}{P}$$
(R) such that {p(x), q(x), r(x)} is an orthogonal set.
1:Find the determinant of the following mattrix $$[((2,-1,-6)),((-3,0,5)),((4,3,0))]$$ 2: If told that matrix A is singular Matrix find the possible value(s) for x $$A = { (16x, 4x),(x,9):}$$
Let T be the linear transformation from R2 to R2 consisting of reflection in the y-axis. Let S be the linear transformation from R2 to R2 consisting of clockwise rotation of 30◦. (b) Find the standard matrix of T , [T ]. If you are not sure what this is, see p. 216 and more generally section 3.6 of your text. Do that before you go looking for help!
Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same $$\displaystyle{P}{\left({x}\right)}={3}{x}^{{4}}-{5}{x}^{{3}}+{x}^{{2}}-{3}{x}+{5}{N}{S}{K}{Q}{\left({x}\right)}={\left({\left({\left({3}{x}-{5}\right)}{x}+{1}\right)}{x}-{3}\right)}{x}+{5}$$ Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial $$\displaystyle{R}{\left({x}\right)}={x}^{{5}}—{2}{x}^{{4}}+{3}{x}^{{3}}—{2}{x}^{{3}}+{3}{x}+{4}$$ in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?
DISCOVER: Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same $$P(x) = 3x^{4} - 5x^{3} + x^{2} - 3x +5$$
$$Q(x) = (((3x - 5)x + 1)x 3)x + 5$$
Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial
R(x) =x^{5} - 2x^{4} + 3x^{3} - 2x^{2} + 3x + 4\) in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head.
Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?
Let $$A = (1, 1, 1, 0), B = (-1, 0, 1, 1,), C = (3, 2, -1, 1)$$
and let $$D = \{Q \in R^{4} | Q \perp A, Q \perp B, Q \perp C\}$$.
Convince me that D is a subspace of $$R^{4}. Write D as span of a basis. Write D as a span of an orthogonal basis. asked 2021-02-12 Write the given matrix equation as a system of linear equations without matrices. \([(2,0,-1),(0,3,0),(1,1,0)][(x),(y),(z)]=[(6),(9),(5)]$$
Let D be the diagonal subset $$\displaystyle{D}={\left\lbrace{\left({x},{x}\right)}{\mid}{x}∈{S}_{{3}}\right\rbrace}$$ of the direct product S_3 × S_3. Prove that D is a subgroup of S_3 × S_3 but not a normal subgroup.
Let the vector space $$P^{2}$$
have the inner product $$\langle p,q\rangle=\int_{-1}^{1} p(x)q(x)dx.$$
Find the following for $$p = 1\ and\ q = x^{2}.$$
$$(a) ⟨p,q⟩ (b) ∥p∥ (c) ∥q∥ (d) d(p,q)$$