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

In summary, the matrix representing the linear map $T$ is $\begin{bmatrix}2&-\tfrac12 & \tfrac43 & -\tfrac34\end{bmatrix}$ with respect to the standard bases of $R[x]_3$ and $R$, and the basis for its kernel is $\{1+4x,2-3x^2,3+8x^3\}$.
  • #1
Mathick
23
0
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.
 
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  • #2
Mathick said:
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 $T(p(x)) = \begin{bmatrix}2&-\tfrac12 & \tfrac43 & -\tfrac34\end{bmatrix} \begin{bmatrix}0\\1\\0\\0\end{bmatrix} = -\frac12.$
 

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

1. What is a matrix of transformation?

A matrix of transformation is a mathematical tool used to represent a geometric transformation in space. It is a square matrix that contains information about the translation, rotation, scaling, and shearing of points in a coordinate system.

2. How is a matrix of transformation used?

A matrix of transformation is used to perform geometric operations on objects. By multiplying a point's coordinates with the transformation matrix, the point is transformed to a new location in space. This is commonly used in computer graphics and computer-aided design.

3. What are the components of a matrix of transformation?

A matrix of transformation contains values for translation in the first three columns, rotation in the next three columns, and scaling in the final three columns. The last row is typically [0 0 0 1] to maintain homogenous coordinates.

4. How does matrix multiplication work with transformation matrices?

To apply multiple transformations to a point, the transformation matrices are multiplied in the order in which they are to be applied. This is known as matrix concatenation. The resulting matrix is then multiplied with the point's coordinates to get the final transformed point.

5. Can a matrix of transformation represent any type of transformation?

Yes, a matrix of transformation can represent any linear transformation in three-dimensional space. However, it cannot represent non-linear transformations such as bending or twisting of objects.

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