Bases

Section 10.3 Bases

A basis for the vector space \(V\) is a set of vectors \(\{v_1, v_2,\dots ,v_k\}\) that is linearly independent and spans \(V\text{.}\) We say that the size of this set is the dimension of \(V\text{.}\)

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Sage implicitly computes the basis and dimension of any subspace generated with the span command.

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Observe that the basis is given in a matrix form. We can explicitly ask for the same result by applying the method basis_matrix.

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With the method basis we obtain the basis as a list of vectors. Whereas the method gens returns the same basis as tuple.

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The dimension command simply returns the number of vectors in the basis.

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Alternatively, we can create the same subspace using the subspace method:

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We can check that we obtain the same vector space.

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Sage row-reduces your input vectors to produce the basis in echelon form. If we want to create a subspace with a specific basis, we can use the subspace_with_basis method. This method takes as input a list of vectors that form a basis of the subspace.

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This method returns an error if we input a list that is not a basis for \(S\text{.}\)

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Observe that we still obtain the same vector space, only the specified basis changes:

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Subsection 10.3.1 Extracting a Basis from a Generator

Every spanning set in a vector space can be reduced to a basis of that vector space. We can manually extract a basis from a given generating set \(\{v_1, v_2, \dots ,v_n\}\) as follows:

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Construct a matrix with \(v_1, v_2, \dots ,v_n\) as its columns.
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Find the pivot columns.
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The columns corresponding to the pivots form a basis of the subspace spanned by the vectors.

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For example, let’s extract a basis of \(S\) from the generator \(v_1,v_2,v_3,v_4\text{.}\)

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We obtain that the pivot columns are the first and the second. Then the vectors \(v_1\) and \(v_2\) forms a basis of \(S\text{.}\) Since we already know that the dimension of \(S\) is 2, this result is consistent. We can also verify that these vectors are linearly independent:

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Subsection 10.3.2 Coordinates in a Basis

If \(B = \{b_1,b_2, \dots, b_n\}\) is a basis of the vector space \(V\text{,}\) then every \(v \in V\) can be expressed uniquely as a linear combination:

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\begin{gather*} v = c_1 b_1 + c_2 b_2 + \dots + c_n b_n \end{gather*}

The scalars \(c_1, c_2, \dots, c_n\) are the coordinates of \(v\) with respect to the basis \(B\) and we write \([v]_B = (c_1, c_2, \dots , c_n)\text{.}\)

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Sage provides dedicated methods to calculate the coordinates of a given vector in any basis. To calculate the coordinates in the canonical echelonized basis of the subspace \(S\text{,}\) we simply use the method coordinates and Sage will return the coordinates as a list. For instance, let’s calculate the coordinates of \(v_1\) in the canonical echelonized basis \(B\) of the subspace \(S\text{.}\) Recall that the canonical basis is given as a list:

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\begin{equation*} B = \{b_1, b_2\} = \{(1,0,1,-1), (0,1,-1/2,1)\} \end{equation*}

Then the coordinates of \(v_1\) are:

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We obtained that the vector \(v_1 = b_1 + 2 \cdot b_2\text{.}\) We can check that this is correct by manually calculating this linear combination and comparing it with the given vector:

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By using the method coordinate_vector we obtain the same coordinates as a vector:

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Note that Sage stores the basis vectors of \(S\) as rows in the basis matrix. To use the usual column-vector convention, we transpose this matrix so that its columns are the basis vectors of \(S\text{.}\) This results in \(M_B \cdot [v_1]_B = v_1\text{,}\) where \(M_B\) is the canonical basis matrix of \(S\text{.}\)

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Subsection 10.3.3 Change of Basis

To calculate the coordinates in any other user defined basis, we use a combination of the methods subspace_with_basis and coordinates. For instance, let’s calculate the coordinates of \(v_1\) in the basis \(C = \{u_1,u_2\}\) of \(S\text{:}\)

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We obtained that the vector \(v_1 = 1/3 \cdot u_1 + 1/3 \cdot u_2\text{.}\) Once again, we can verify it by manually calculating this linear combination and comparing it with the given vector:

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As expected, when we compute the coordinates of the basis vectors themselves, we obtain the standard unit vector coordinates.

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Subsection 10.3.4 Change of Basis Matrix

Often we need to convert the coordinates of a vector from one basis \(B\) to another basis \(C\text{.}\) This is done using a change-of-basis matrix, \(P_{C \leftarrow B}\text{.}\) The conversion formula is:

\begin{equation*} [v]_C = P_{C \leftarrow B} \cdot [v]_B \end{equation*}

where \([v]_B\) and \([v]_C\) are column vectors of coordinates. The columns of the matrix \(P_{C \leftarrow B}\) are the coordinate vectors of the vectors in basis \(B\) expressed in terms of basis \(C\text{.}\)

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Let’s calculate the transition matrix from the canonical basis \(B\) of \(S\) to the basis \(C = \{u_1, u_2\}\) of \(T\text{.}\)

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We will first define our initial basis \(B\text{.}\)

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We will now produce our change of basis matrix by using the coordinates of the vectors in \(B\) with respect to the basis \(C\text{.}\)

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Now we can use this matrix to convert the coordinates of \(v_1\) from basis \(B\) to basis \(C\text{.}\)

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We can verify that this matches the coordinates we calculated earlier using the coordinates method:

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