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 : [math]R[x]_3 →R[/math] be the function defined by [math] T(p(x)) = p(−1) + \int_{0}^{1} p(x) \,dx [/math], where [math]R[x]_3[/math] 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 [math]R[x]_3[/math] and [math]R[/math] (the latter basis is of course given by [math] \left\{(1)\right\}[/math], 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 [math]A=(2 \quad -\frac{1}{2} \quad\quad \frac{4}{3} \quad -\frac{3}{4})[/math]. And then the kernel basis is [math]1+4x,2-3x^2,3+8x^3[/math].

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

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 : [math]R[x]_3 →R[/math] be the function defined by [math] T(p(x)) = p(−1) + \int_{0}^{1} p(x) \,dx [/math], where [math]R[x]_3[/math] 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 [math]R[x]_3[/math] and [math]R[/math] (the latter basis is of course given by [math] \left\{(1)\right\}[/math], 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 [math]A=(2 \quad -\frac{1}{2} \quad\quad \frac{4}{3} \quad -\frac{3}{4})[/math]. And then the kernel basis is [math]1+4x,2-3x^2,3+8x^3[/math].

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

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 $T(p(x)) = \begin{bmatrix}2&-\tfrac12 & \tfrac43 & -\tfrac34\end{bmatrix} \begin{bmatrix}0\\1\\0\\0\end{bmatrix} = -\frac12.$