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LINEAR INDEPENDENCE AND BASES[4.3]
Recall: Linear independence
Definition ä The set {v 1 , ..., vp} is said to be linearly dependent if there exist weights c 1 , ..., cp, not all zero, such that c 1 v 1 + c 2 v 2 + ... + cpvp = 0 (1) ä It is linearly independent otherwise ä The above equation is called linear dependence relation among the vectors v 1 , · · · , vp
ä The set v 1 , v 2 , · · · , vp is linearly dependent if and only if the equation (1) has a nontrivial solution, i.e., if there are some weights, c 1 , ..., cp, not all zero, such that (1) holds. ä In such a case, (1) is called a linear dependence relation among v 1 , ..., vp.
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Theorem: An indexed set {v 1 , ..., vp} of two or more vec- tors, with v 1 6 = 0, is linearly dependent if and only if some vj (with j > 1 ) is a linear combination of the preceding vectors, v 1 , ..., vj− 1.
- As an exercise prove formally this theorem Definition: Let H be a subspace of a vector space V. An indexed set of vectors B = {b 1 , ..., bp} in V is a basis for H if:
- B is a linearly independent set, and
- The subspace spanned by B coincides with H; that is, H = span{b 1 , ..., bp}
ä The definition of a basis applies to the case when H = V , (any vector space is a subspace of itself) ä A basis of V is a linearly independent set that spans V. ä Note that condition (2) implies that each of the vec- tors b 1 , ..., bp must belong to H, because span{b 1 , ..., bp} contains b 1 , ..., bp.
Standard basis of Rn
Let e 1 , ..., en be the columns of the n × n matrix, In.
That is,
e 1 =
; e 2 =
; · · · ; en =
ä The set {e 1 , · · · , en} is called the standard basis for Rn. ä Sometimes the term canonical basis is used
x
x 1
2
x 3
e
e 1 e^2
3
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Spanning set theorem Theorem: Let S = {v 1 , ..., vp} be a set in V , and let H = span{v 1 , ..., vp}.
- If one of the vectors in S–say, vk–is a linear combination of the remaining vectors in S, then the set formed from S by removing vk still spans H.
- If H 6 = { 0 }, some subset of S is a basis for H.
Proof: 1. By rearranging the list of vectors in S, if neces- sary, we may assume that vk is the last vector of the list, i.e., vp, so: vp = a 1 v 1 + ... + ap− 1 vp− 1 (1)
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ä Given any x in H, we may write
x = c 1 v 1 + ... + cp− 1 vp− 1 + cpvp (2)
for suitable scalars c 1 , ..., cp.
ä Substituting the expression for vp from (1) into (2) it is easy to see that x is a linear combination of v 1 , ...vp− 1.
ä Vector x was arbitrary – Thus {v 1 , ..., vp− 1 } spans H -
- If the original spanning set S is linearly independent, then it is already a basis for H.
ä Otherwise, one of the vectors in S depends on the others and can be deleted, by part (1).
ä Repeat this process until the spanning set is linearly independent and hence is a basis for H.
ä If the spanning set is eventually reduced to one vector, that vector will be nonzero (and hence linearly indepen- dent) because H 6 = { 0 }.
- Let H = span{v 1 , v 2 , v 3 } with
v 1 =
;^ v 2 =
;^ v 3 =
Show that v 3 is a linear combination of the first 2 vectors and then find a basis of H.
ä Related (and important) definition
Definition: The rank of a matrix A is the dimension of its column space.
ä Notation: rank(A).
ä Note: rank(A) = number of pivot columns in A.
ä Recall from an earlier example that we could find a spanning set of Nul(A) which has as many vectors as there are free variables.
ä Therefore dim(Nul(A)) = number of free variables. Hence the important result rank(A) + dim(Nul(A)) = n
ä Known as the Rank+Nullity theorem
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APPLICATION: ROTATION AND TRANSLATIONS [2.7]
Application: Rotations and translations in R^2
ä In the form of exercises. Try to answer all questions before class [see textbook if needed]
- Consider the mapping that sends any point x in R^2 into a point y in R^2 that is rotated from x by an angle θ. Is the mapping linear?
x
x
45 deg.
- Find the matrix representing the mapping. [Hint: ob- serve how the canonical basis is transformed]
Rotations and translations in R^2
ä We will now deal with Translations or shifts ä Another very important operation.. ä Recall: Not a linear mapping – but called affine map- ping.. ä This will require a little artifice..
- How can you now represent a translation via a matrix- vector product? [Hint: add an artificial compoment of 1 at the end of vector x] ä Called Homogeneous coordinates ä Try this in matlab
Rotations and translations in R^2
- The most important mapping in real life is a combination of Rotation and Translation. How do you represent these?
ä We will use the Homogeneous coordinates introduced above
ä Need to combine two mappings: rotation and then translation
- Does the order matter? Reason from the geometry and then from the derivation of your matrix
- Find the combined mapping
ä Try this in matlab
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