Linear Algebra – Real Euclidean Vector Spaces - Quick Guides

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Study GuideLinear Algebra–Real Euclidean Vector Spaces1.Linear Combinations and SpanWhat Is a Linear Combination?Let’s begin with the basics.Suppose we have vectorsin (Rn).Alinear combinationof these vectors means we multiply each vector by a number (called a scalar)and then add them together.It looks like this:Here:•(k1, k2,. . ., kr) arescalars(real numbers).•The vectors are combined using multiplication and addition.Insimple words:A linear combination is just a “weighted sum” of vectors.Example 1:A Simple Linear CombinationLet’s look at a concrete example.

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Study GuideWe are given:Now consider the vector:It turns out:That means we multiplied (v1) by 2,multiplied (v2) by–3, and added them together. So (v) is alinearcombinationof (v1) and (v2).Important IdeaThezero vectoris always a linear combination of any vectors.Why? Because:In fact, the zero vector in (Rn) can always be written as a linear combination of any collection ofvectors.What Is the Span?Now that we understand linear combinations, let’s go one step further.Thespanof a set of vectors is:The set ofall possible linear combinationsof those vectors.If we havevectors:

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Study GuideThen their span is written as:Why Is Span Important?The span:•Always forms asubspaceof (Rn)•Is closed under addition•Is closed under scalar multiplicationThat means if you add two vectors from the span, or multiply one by ascalar, you stay inside thespan.If a subspace (V) equalsExample 2:Span in(R3)Now let’s move to three dimensions.Consider the vectors:Their span consists of all linear combinations of these two vectors.Since we have two vectors in (R3), their span forms aplanein three-dimensional space.

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Study GuideFinding the Plane EquationTo find the equation of this plane, we calculate anormal vectorusing the cross product:Using this normal vector, the plane equation has the form:Because the span mustcontain the origin (it’s a subspace), we substitute (0,0,0):So the plane is:This plane passes through the origin.Example 3:A Very Important CaseNow consider the standard unit vectors in (R2):Every vector in (R2) can be written as:So:This means:

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Study GuideThe vectors (i) and (j) generate the entire plane.They form aspanning setfor (R2).Removing Unnecessary VectorsHere’s an important idea for efficiency:If one vector in a collection can be written as a linear combination of the others, then it isnot neededto form the span.In that case:So we can safely remove the dependent vector.Example 4:When One Vector Depends on the OthersSuppose we have:We are told that:What Does This Mean?This means(v3) is a linear combination of (v1) and (v2).So (v3) doesn’t add anything new to the span. It is already built from the first twovectors.Therefore:

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Study GuideWhy Can We Remove (v3)?Since (v3) can be written using (v1) and (v2), it lies in thesame planeformed by those two vectors.Geometrically:•(v1) and (v2) span a plane in (R3)•(v3) sits on that same plane•Addingmultiples of (v3) won’t create anything outside that planeSo removing (v3) does not change the span.Even More Interesting…It turns out the dependence goes both ways!So each vector can be written using the other two.That means:Any one of thethree vectors can be removed without changing the span.This is a perfect example oflinear dependence.

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Study GuideExample 5:When a Vector Does Add Something NewNow consider a slightly different situation:This time, there areno scalars(k1) and (k2) such that:So (v3) isnota linear combination of (v1) and (v2).What Does That Mean Geometrically?We already know:•(v1) and (v2) span a plane in (R3)But now:•(v3) doesnotlie in that plane•It points in a new directionSo adding (v3) allows us to reach vectorsoutside the plane.Figure 1Final ConclusionBecausev3is not in the plane spanned byv1andv2

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Study Guidecontains vectors that arenotin:In fact, together the three vectors spanall ofR3.2. Linear IndependenceWhatDoes Linear Dependence Mean?be a collection of vectors in (Rn), where(r > 2).We say the vectors arelinearly dependentif at least one vector in the collection can be written as alinear combination of the others.In simple words:•If one vector “depends on” the others, the set isdependent.•If no vector can be built from the others, the set isindependent.It is common to say:“The vectors are linearly independent”instead of“The set containing these vectors is linearly independent.”Both statements mean the same thing.An Equivalent (and Often Easier) Definition

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Study GuideThere is another definition that iscompletely equivalent but often simpler to use.arelinearly independentif the only scalars that satisfyareThis is called thetrivial solution.If there exists any solution where the scalars are not all zero (called anontrivial solution), then thevectors arelinearly dependent.Example 1Are the vectorslinearly independent?Step 1:Assume One Vector Depends on the OthersTo test independence, we check whether one vector can be written as a combination of the other two.Suppose (v3) can bewritten as:Substituting the vectors:

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Study GuideThis gives the system of equations:Step 2:Solve the SystemSubtract the first equation from the third:This gives:Substitute back into the first equation:Now check the second equation:But the equation requires it to equal−2, not −8.So the system isinconsistent.Conclusion of Example 1

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