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Linear Algebra – Real Euclidean Vector Spaces

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    Study Guide Linear Algebra – Real Euclidean Vector Spaces 1. Linear Combinations and Span What Is a Linear Combination? Let’s begin with the basics. Suppose we have vectors in (R n ). A linear combination of these vectors means we multiply each vector by a number (called a scalar) and then add them together. It looks like this: Here: • (k 1 , k 2 , . . . , k r ) are scalars (real numbers). • The vectors are combined using multiplication and addition. In simple words: A linear combination is just a “weighted sum” of vectors. Example 1: A Simple Linear Combination Let’s look at a concrete example.

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    Study Guide We are given: Now consider the vector: It turns out: That means we multiplied (v 1 ) by 2, multiplied (v 2 ) by – 3, and added them together. So (v) is a linear combination of (v 1 ) and (v 2 ). Important Idea The zero vector is always a linear combination of any vectors. Why? Because: In fact, the zero vector in ( R n ) can always be written as a linear combination of any collection of vectors. What Is the Span? Now that we understand linear combinations, let’s go one step further. The span of a set of vectors is: The set of all possible linear combinations of those vectors. If we have vectors:

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    Study Guide Then their span is written as: Why Is Span Important? The span: • Always forms a subspace of (R n ) • Is closed under addition • Is closed under scalar multiplication That means if you add two vectors from the span, or multiply one by a scalar, you stay inside the span. If a subspace (V) equals Example 2: Span in (R 3 ) 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 (R 3 ), their span forms a plane in three - dimensional space.

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    Study Guide Finding the Plane Equation To find the equation of this plane, we calculate a normal vector using the cross product: Using this normal vector, the plane equation has the form: Because the span must contain 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 Case Now consider the standard unit vectors in (R 2 ): Every vector in (R 2 ) can be written as: So: This means:

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    Study Guide The vectors (i) and (j) generate the entire plane. They form a spanning set for (R 2 ). Removing Unnecessary Vectors Here’s an important idea for efficiency: If one vector in a collection can be written as a linear combination of the others, then it is not needed to form the span. In that case: So we can safely remove the dependent vector. Example 4: When One Vector Depends on the Others Suppose we have: We are told that: What Does This Mean? This means (v 3 ) is a linear combination of (v 1 ) and (v 2 ) . So (v 3 ) doesn’t add anything new to the span. It is already built from the first two vectors. Therefore:

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    Study Guide Why Can We Remove (v 3 )? Since (v 3 ) can be written using (v 1 ) and (v 2 ), it lies in the same plane formed by those two vectors. Geometrically: • (v 1 ) and (v 2 ) span a plane in (R 3 ) • (v 3 ) sits on that same plane • Adding multiples of (v 3 ) won’t create anything outside that plane So removing (v 3 ) 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 the three vectors can be removed without changing the span. This is a perfect example of linear dependence .

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    Study Guide Example 5: When a Vector Does Add Something New Now consider a slightly different situation: This time, there are no scalars (k 1 ) and (k 2 ) such that: So (v 3 ) is not a linear combination of (v 1 ) and (v 2 ). What Does That Mean Geometrically? We already know: • (v 1 ) and (v 2 ) span a plane in (R 3 ) But now: • (v 3 ) does not lie in that plane • It points in a new direction So adding (v 3 ) allows us to reach vectors outside the plane . Figure 1 Final Conclusion Because v 3 is not in the plane spanned by v 1 and v 2

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    Study Guide contains vectors that are not in: In fact, together the three vectors span all of R 3 . 2. Linear Independence What Does Linear Dependence Mean? be a collection of vectors in (R n ), where ( r > 2 ) . We say the vectors are linearly dependent if at least one vector in the collection can be written as a linear combination of the others. In simple words: • If one vector “depends on” the others, the set is dependent . • If no vector can be built from the others, the set is independent . 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 Guide There is another definition that is completely equivalent but often simpler to use. are linearly independent if the only scalars that satisfy are This is called the trivial solution . If there exists any solution where the scalars are not all zero (called a nontrivial solution ), then the vectors are linearly dependent . Example 1 Are the vectors linearly independent? Step 1: Assume One Vector Depends on the Others To test independence, we check whether one vector can be written as a combination of the other two. Suppose (v 3 ) can be written as: Substituting the vectors:

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    Study Guide This gives the system of equations: Step 2: Solve the System Subtract 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 is inconsistent . Conclusion of Example 1

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