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Linear Algebra: A second course, featuring proofs and Python

Sean Fitzpatrick

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Worksheet 1.5 Worksheet: working with span
A4US

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In this worksheet, we will attempt to understand the concept of span. Recall from Section 1.4 that the span of a set of vectors v1,v2,…,vk in a vector space V is the set of all linear combinations that can be generated from those vectors.
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Recall the following fact: given a matrix A and vector x, Ax is a linear combination of the columns of A.
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In particular, if we write A in terms of its columns as A=[v1v2⋯vn] then
Ax=x1v1+x2v2+⋯+xnvn,
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where x1,x2,…,xn are the components of x.
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Let’s confirm this for the matrix A=[23−56−720−11643−6312] and vector x=[7−9314].
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Input the matrix and vector in the cell below.
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See Section B.3 for details on how to enter your matrix and vector, and how to compute the following.
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Next, compute Ax:
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Finally, compute the corresponding linear combination of the columns of A, and confirm that you get the same answer.
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The question “Does the vector w belong to the span of v1,v2,…,vk?” is equivalent to asking, “Can I write w as a linear combination of the vi?” This, in turn, is equivalent to asking:
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Do there exist scalars x1,x2,…,xk such that
w=x1v1+x2v2+⋯+xkvk?
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If the above vectors are vectors in Rn, then we just saw that the right-hand side of the above equation is Ax, where v1,v2,…,vk are the columns of A. So the question of whether w is in the span of the vi is equivalent to determining whether the equation Ax=w has a solution.
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1.

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Determine whether or not the vector w=⟨3,−1,4,2⟩ in R4 belongs to the span of the vectors
⟨2,1,4,−3⟩,⟨0,2,1,4⟩,⟨−1,1,0,2⟩.
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To assist with solving this problem, a code cell is provided below. Once you have determined the augmented matrix of your system of equations, enter the matrix, and then compute its reduced row-echelon form.
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(Do you recall how to tell if a system has a solution from the RREF?)
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If our vector space is not Rn, we can still apply the above technique, but we first have to turn our span equation into a system of linear equations.
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2.

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Determine whether or not the polynomial q(x)=4−6x−11x2 belongs to the span of the polynomials
p1(x)=x−3x2,p2(x)=2−x,p3(x)=−1+4x+x2.
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3.

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Determine whether or not the matrix B=[4−17−562] belongs to the span of the matrices
A1=[120−324],A2=[32−7013],A3=[01320−4],A4=[−113201].
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The above problems all answer the same basic question: given a particular vector w, is it in the span of given vectors v1,v2,…,vk?
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A more interesting problem is the following: iven the vectors v1,v2,…,vk, what are all the vectors in their span? That is, how can we describe the span of a set of vectors more directly?
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We can answer this question as follows: let w be an arbitrary vector in V, and write it in terms of its (unknown) coefficients. For example, in question 1 above, we might let w=⟨a,b,c,d⟩.
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4.

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For each problem above, define symbolic variables using the symobols command. For example, you can enter
a, b, c = symbols('a b c')
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to create three undetermined symbols.
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Then replace the last column in each augmeted matrix from the previous problems with a column of symbols, and instead of using the rref() command, use the echelon_form() command to reduce the matrix.
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By looking at the reduced form of hte matrix, determine a relationship (or relationships) between the variables that must hold for the system to be consistent. You can then use this to define the set of vectors in the span.
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