# injective but not surjective linear map

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jan 8, 2021

1 If dim(V) >dim(W), then T is not injective. Recall that the composition TS is defined by (TS)(x) = T(S(x)). Injective, Surjective and Bijective "Injective, Surjective and Bijective" tells us about how a function behaves. Let $$T : V \rightarrow W$$ be a linear map between vector spaces. An injective map between two finite sets with the same cardinality is surjective. then a linear map T : V !W is injective if and only if it is surjective. Linear algebra An injective linear map between two finite dimensional vector spaces of the same dimension is surjective. (a) Prove that if TS is injective, then S is injective. So this would be a case where we don't have a surjective function. But this would still be an injective function as long as every x gets mapped to a unique y. Proof. He doesn't get mapped to. This is not onto because this guy, he's a member of the co-domain, but he's not a member of the image or the range. 2 If dim(V) V and T: V -> W be two linear maps. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 10 / 1 Then: In particular, ker(T) = f0gif and only if T is bijective. A function $f: R \rightarrow S$ is simply a unique “mapping” of elements in the set $R$ to elements in the set $S$. Surjective Linear Map Corollary Let T : V !W be a linear map. However, if we restrict ourselves to … Example 5. Deﬁnition 5. Let F be a linear map from R3 to R3 such that F (x, y, z) = (2x, 4x − y, 2x + 3y − z), then a) F is injective but not surjective b) F is surjective but not injective c) F is neither injective nor surjective d) … (b) Prove that if TS is surjective, then T is surjective. By the rank-nullity theorem, the dimension of the kernel plus the dimension of the image is the common dimension of V and W, say n. By the last result, T is injective Lemma 3.6.2. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … A linear map T : V → W is called surjective if rangeT = W. A linear map T : V → W is called bijective if T is injective and surjective. A function is a way of matching the members of a set "A" to a set "B": Let's look at that more closely: A General Function points from each member of "A" to a member of "B".