# 线性代数代写|Linear Algebra I MA1008 Cardiff University Assignment

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## Instructions:

Linear algebra is a branch of mathematics that deals with the study of vector spaces, linear transformations, and matrices. It is widely used in various fields of science and engineering, including physics, computer science, and economics.

The concept of a vector space is fundamental to linear algebra. A vector space is a collection of vectors that satisfy certain axioms, such as closure under addition and scalar multiplication. Examples of vector spaces include the set of real numbers, the set of polynomials, and the set of functions.

Subspaces are subsets of vector spaces that are also vector spaces in their own right. They are important because they allow us to focus on smaller, more manageable subsets of a larger vector space.

Linear transformations are functions that preserve the structure of a vector space. They are also known as linear maps or linear operators. Examples of linear transformations include rotations, reflections, and scaling.

Linear combinations are a way of combining vectors using scalar multiplication and addition. Spanning sets are sets of vectors that can be used to generate all the vectors in a vector space through linear combinations. Linearly independent sets are sets of vectors that cannot be generated through linear combinations of other vectors in the set.

Dimensionality is a measure of the size of a vector space. It is defined as the number of vectors in a basis of the vector space. A basis is a set of linearly independent vectors that span the vector space.

The final result that every finite-dimensional vector space can be identified with a set of finite-tuples of scalars is known as the coordinate representation theorem. It states that any vector in a finite-dimensional vector space can be uniquely represented by a finite sequence of scalars, which can be thought of as coordinates.

Overall, linear algebra is a powerful tool that enables us to study the properties of vectors and linear transformations in a systematic and rigorous way. It has many applications in various areas of mathematics and beyond.

Suppose $A v_i=b_i$ for the vectors $v_1, \ldots, v_n$ and $b_1, \ldots, b_n$ in $R^n$. Put the $v$ ‘s into the columns of $V$ and put the $b$ ‘s into the columns of $B$.
(a) Write those equations $A v_i=b_i$ in matrix form. What condition on which vectors allows $A$ to be determined uniquely? Assuming this condition, find $A$ from $V$ and $B$.

$A\left[v_1 \cdots v_n\right]=\left[b_1 \cdots b_n\right]$ or $A V=B$. Then $A=B V^{-1}$ if the $v^{\prime} s$ are independent.

(b) Describe the column space of that matrix $A$ in terms of the given vectors.

The column space of $A$ consists of all linear combinations of $b_1, \cdots, b_n$.

(c) What additional condition on which vectors makes $A$ an invertible matrix? Assuming this, find $A^{-1}$ from $V$ and $B$.

If the $b^{\prime} s$ are independent, then $B$ is invertible and $A^{-1}=V B^{-1}$.