Why Does the Matrix Calculation Not Match Expected Results in Linear Mapping?

[Mathick]

Hello!

I don't know exactly how to state my question so I'll show you what my problem is.

Ex. Let T : \(\displaystyle R[x]_3 →R\) be the function defined by \(\displaystyle T(p(x)) = p(−1) + \int_{0}^{1} p(x) \,dx \), where \(\displaystyle R[x]_3\) is a vector space of polynomials with degree at most 3. Show that $T$ is a linear map; write down the matrix of $T$ with respect to the standard bases of \(\displaystyle R[x]_3\) and \(\displaystyle R\) (the latter basis is of course given by \(\displaystyle \left\{(1)\right\}\), i.e. the set consisting of the column vector of length 1 with entry 1), and find a basis for the kernel of $T$.

I found this matrix. It's a row vector \(\displaystyle A=(2 \quad -\frac{1}{2} \quad\quad \frac{4}{3} \quad -\frac{3}{4})\). And then the kernel basis is \(\displaystyle 1+4x,2-3x^2,3+8x^3\).

But I don't understand it. I mean if I multiply the matrix \(\displaystyle A\) by a standard vector \(\displaystyle (1 \quad x \quad x^2 \quad x^3) \) I won't get, for example, for \(\displaystyle p(x)=x \) the value \(\displaystyle -\frac{1}{2}\).

Please, help me find the point where I misunderstand something. I really want to know what I am doing. I don't want to do maths like a robot without understanding.

 

[Opalg]

Hello!

I don't know exactly how to state my question so I'll show you what my problem is.

Ex. Let T : \(\displaystyle R[x]_3 →R\) be the function defined by \(\displaystyle T(p(x)) = p(−1) + \int_{0}^{1} p(x) \,dx \), where \(\displaystyle R[x]_3\) is a vector space of polynomials with degree at most 3. Show that $T$ is a linear map; write down the matrix of $T$ with respect to the standard bases of \(\displaystyle R[x]_3\) and \(\displaystyle R\) (the latter basis is of course given by \(\displaystyle \left\{(1)\right\}\), i.e. the set consisting of the column vector of length 1 with entry 1), and find a basis for the kernel of $T$.

I found this matrix. It's a row vector \(\displaystyle A=(2 \quad -\frac{1}{2} \quad\quad \frac{4}{3} \quad -\frac{3}{4})\). And then the kernel basis is \(\displaystyle 1+4x,2-3x^2,3+8x^3\).

But I don't understand it. I mean if I multiply the matrix \(\displaystyle A\) by a standard vector \(\displaystyle (1 \quad x \quad x^2 \quad x^3) \) I won't get, for example, for \(\displaystyle p(x)=x \) the value \(\displaystyle -\frac{1}{2}\).

Please, help me find the point where I misunderstand something. I really want to know what I am doing. I don't want to do maths like a robot without understanding.

With respect to the standard basis $\{1,x,x^2,x^3\}$ of $R[x]_3$, the polynomial $p(x) = a + bx + cx^2 + dx^3$ is represented by the vector $V = \begin{bmatrix}a\\b\\c\\d\end{bmatrix},$ so that $$T(p(x)) = AV = \begin{bmatrix}2&-\tfrac12 & \tfrac43 & -\tfrac34\end{bmatrix} \begin{bmatrix}a\\b\\c\\d\end{bmatrix}.$$ In particular, if $p(x) = x$ then $V = \begin{bmatrix}0\\1\\0\\0\end{bmatrix}$ and $T(p(x)) = \begin{bmatrix}2&-\tfrac12 & \tfrac43 & -\tfrac34\end{bmatrix} \begin{bmatrix}0\\1\\0\\0\end{bmatrix} = -\frac12.$