Quiz
Quiz yourself by thinking what should be in
each of the black spaces below before clicking
on it to display the answer.
Help!
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| Eight rules for a linear space or vector space | associative, commutative, neutral element,has negative, distributive, associative with products, same as mulitple of 1
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| A linear space is | a set with two reasonably defined operations, addition and scalar multiplication, that allow us to form linear combinations.
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| The neutral element for an nxm matrix | the 0 matrix
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| The neutral elemtn for the linear equations in three unknowns: ax+by+cz=d | is a=b=c=d=0
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| The neutral element of a vector | the 0 vector
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| K(a+ib)=ka+i(kb) form a linear space and have the neutral element | 0=0+0i
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| Subspaces: A subset W of a linear space V is called a subspace of V if | a. contains the neutral element 0 of V
b. is closed under addition
c. W is closed under scalar multiplication
(b/c is closed under linear combinations)
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| Differentiable functions form a subspace, True or False? | True
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| Examples of subspaces | c^(infinity), P the set of all polynomials, Pn the set of all polynomials of decree less than or equal to n
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| we say that f1....f2 span V if | every f in V can be expressed as a linear combination of f1...f2
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| We say that fi is redundant if | it is a linear combinationf f1...fi-1. The elements are called linearly independent if none of them is redundant.
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| If equation c1f1+...+cnfn=0 | has only the trivial solution c1=...=cn=0
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| We say that f1...fn are a basis of V if they | span V and are linearly independent. This means that every f in V can be written uniquely as a linear combination.
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| The coefficients c1...cn are called the | coordinates of f with respect to the basis B=(f1...fn)
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| B-coordinate transformation is invertible 1 | True
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| [f+g]B=[g]B+[f]B T/F | T
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| Finding a basis of a linear space | a. Write down a typical element of V, int terms of some arbitrary constants
b. Use arbitrary constants as coeff, express typical element as a linear combination of some elements of V
c. Verify the elements are linearly independent
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| Finite dimensional linear spaces | A linear space V is called finite dimensional if it has a (finite)basis so that we can define its demenension dim(V)=n
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| The image is a subspace of the and the kernel is a subspace of the | target-domain
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| If the image of T is finite dimensional, then dim(imT) is called the | rank of T
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| If the kernel of T is finite dimensional, then dim(kerT) is called | nullity of T
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| dim(V)= | rank(T) +nullity(T)
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| dim(V)= | dim(imT)+dim(kerT)
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| Show linear transformation | show that closed under scalar multiplication and addition
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| The kernel consists of | all inputs into T that produce 0 as output
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| Is T 1-1 check | Ker(T) must =0
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| 2out of 3 Rule | -dim(V)=dim(W)
-ker(T)=0 (one-to-one)
-im(T)=W (onto)
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| If one of the three is false then | Not isomorphic
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| If 2 out of three is true | Isomporphic
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| P2-> LB and LC-> R^3 |
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| Sc->B = | (SB->C)^-1
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| Isomorphic | same structure
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