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MATH2940课程简介
Course information provided by the Courses of Study 2022-2023.
Linear algebra and its applications. Topics include: matrices, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality and inner product spaces; applications include brief introductions to difference equations, Markov chains, and systems of linear ordinary differential equations. May include computer use in solving problems.
Prerequisites
When Offered Fall, Spring, Summer.
Prerequisites/Corequisites Prerequisite: MATH 1920.
Forbidden Overlaps Forbidden Overlap: due to an overlap in content, students will receive credit for only one course in the following group: MATH 2210, MATH 2230, MATH 2310, MATH 2940.
Distribution Category (MQR-AS, SMR-AS)
Comments Taking MATH 2930 and MATH 2940 simultaneously is not recommended. For guidance in selecting an appropriate course, please consult First Steps in Math.
MATH2940 linear algebra HELP（EXAM HELP， ONLINE TUTOR）
Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^4$ be the function given by the formula
$$
T\left(x_1, x_2, x_3\right)=\left(x_1+2 x_2-x_3, 3 x_1+6 x_2, x_1+2 x_2, x_2+4 x_3\right)
$$
(a) $T$ is a linear transformation. What is the standard matrix $A$ of $T$ ?
To find the standard matrix A of a linear transformation T, we need to determine the images of the standard basis vectors under T, and then express those images as column vectors in a matrix.
Let {e1, e2, …, en} be the standard basis of the domain of T, and let {v1, v2, …, vm} be the basis of the codomain of T. Then the standard matrix A of T is an m x n matrix whose columns are the coordinates of T(e1), T(e2), …, T(en) with respect to the basis {v1, v2, …, vm}.
That is, if T(ei) = a1i v1 + a2i v2 + … + ami vm, then the ith column of A is [a1i, a2i, …, ami]T.
So, to find the standard matrix A of T, we need to compute T(ei) for each i, express each T(ei) as a linear combination of the basis vectors {v1, v2, …, vm}, and then write the coefficients of each linear combination as a column vector.
For example, consider the linear transformation T: R2 -> R3 defined by T(x, y) = (2x + y, x – y, 3x + 2y). To find the standard matrix A of T, we need to compute T(e1) and T(e2), where e1 = (1, 0) and e2 = (0, 1) are the standard basis vectors of R2:
T(e1) = T(1, 0) = (2, 1, 3) T(e2) = T(0, 1) = (1, -1, 2)
Next, we need to express T(e1) and T(e2) as linear combinations of the standard basis vectors of R3, which are {e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1)}:
T(e1) = 2e1 + e2 + 3e3 T(e2) = e1 – e2 + 2e3
Finally, we write the coefficients of each linear combination as a column vector, and obtain the standard matrix of T:
A = [2 1 1 -1 3 2]
Is the transformation T onto? Justify your answer.
In mathematics, a transformation can refer to different types of functions that map elements from one set to another, such as linear transformations, affine transformations, or continuous transformations.
Whether a transformation is onto or not depends on the specific definition and properties of the transformation in question. In general, a transformation is said to be onto (or surjective) if every element in the codomain (or range) is mapped to by at least one element in the domain. If there exists at least one element in the codomain that is not mapped to by any element in the domain, then the transformation is not onto.
Thus, to determine if a specific transformation T is onto or not, we need to know the definition of T, the domain and codomain of T, and whether all elements in the codomain are mapped to by some element in the domain.
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